Locating waste pipelines to minimize their impact on marine environment

a Locating waste pipelines to minimize their impact on marine environment Teresa Ca ´ceres a , Juan A. Mesa b , Francisco A. Ortega c Department of Applied Mathematics I, Computer Engineering Higher Technical School, University of Seville, Spain b c Department of Applied Mathematics II, Engineering Higher Technical School, University of Seville, Spain Department of Applied Mathematics I, Architecture Higher Technical School, University of Seville, Spain Abstract A waste pipeline, considered as an undesirable facility, is to be located in a coastal region. Two criteria are taken into account, the Euclidean distance from a given set of protected areas (coral reefs and sandbanks) and a utility function related to the pipe length, both to be maximized. The paper describes a methodology to obtain an efficient set of points where the extreme of a marine pipeline should be located. Since the formulation of the model is based on the zone Voronoi diagram, the computational complexity of the solving procedure is low. Keywords: Undesirable facility location; Forbidden regions; Zone Voronoi diagram 1. Introduction The coastal fringe is the territory where marine, air and terrestrial environments interrelate. Here very diverse and fragile ecosystems coexist, although subjected in many cases to increasing degradation due to industrial and urban developments disrespectful to the environment. Nearly two-thirds of the world’s population live along the seaboard. In Spain this proportion can be estimated as 69% by using data corresponding to year 2001 (Instituto Nacional de Estadı ´stica, 2004). Almost a quarter of the Mediterranean littoral in Spain is artificial surface. Most marine contamination is produced on land. Each year, 10 billion tons of industrial and urban sewage are directly poured into the Mediterranean sea, and 90% is untreated. Pollution specifically affects oceanic prairies and coral reefs due to decreasing vegetable biomass and biological diversity (see Worldwide Fund for Nature/Adena, 2000). Posidonia oceanica (Linnaeus) Delile is a plant with leaves, flowers and fruit, similar to those plants which live in forests and gardens, but which lives in the sea between the surface and a depth of 50 m, where there is still enough light for photosynthesis. It is endemic to the Mediterranean sea and, by providing the principal source of oxygenation of the Mediterranean sea, it is its most important ecosystem. The Posidonia meadow produces a barrier reef which maintains the balance of the littoral sedimentation since their long leaves restrain the swell thereby protecting the littoral from erosion by minimizing the impact of the waves on the beaches (see Marba ´et al., 1996). The Posidonia meadow is the habitat of more than 400 plant species and 1000 animal species; it provides shelter, food and an adequate environment for the reproduction of many profitable species. Due to its ecological role, this sea-grass is a protected species in Spain and in France. Nowadays, Posidonia oceanica is decreasing on the coastal fringe due to several factors: 1. Sea contamination, fundamentally from terrestrial sources, produces a subsequent reduction in water quality, which obstructs photosynthesis, thereby causing the death of the plant. 2. Illegal coastal drag fishing is harmful because of its strong physical impact. 3. Public works on the coastal fringe (sports harbours, breakwaters, regeneration of beaches) modify the littoral dynamic and therefore the environmental conditions of the sea floor. 4. The frequent seasonal anchorage of ships in the same places on the coast can damage the sea floor. The European Union (EU) recognizes the need to protect the habitats and marine ecosystems in coastal waters; in particular, meadows of Posidonia oceanica, coral reefs and sandbanks (see EU Directive, 1992). Following the corresponding EU-directives, government agencies are at present promoting the planning and the management of coastal environments. Research carried out in different zones, to determine the distribution and boundaries of the different substratum and sediments of sea-grass beds along the coast, commonly generate a database of visual, topographic and thematic maps so that the analysis by means of a Geographic Information System (GIS) can be performed (see Calzadilla Pe ´rez et al., 2002; Jayatissa et al., 2002; Melloul and Collin, 2002). Once the geographical location of the study area has been established, vigilance and control systems of these habitats are applied in order to detect, by aerial photographs, turbidities of the water column’s vertical structure. For this purpose, the collected satellite images are processed with the GIS in order to assess the interaction between the detected disturbances and the Posidonia oceanica beds. In Fig. 1 a satellite image is shown of the south-west coast of Spain near the mouth of the Tinto river (province of Huelva, Andalusia). Simulation tools, which have been recently developed to assess the water quality, integrate the software to manage teledetection and geo-statistical information. The images generated by those systems are typically planar (see He, 2003). Therefore, the formulation of problems on a plane makes sense although their nature is tridimensional. The setting considered in this paper is a geometrical abstraction of a coastal scenario which consists of a planar region, which includes zones of biological interest, whose left-hand side represents the littoral line. Taking into account the existence of upper and lower boundaries for the township in charge of the installation of the pipeline, the scenario will be assumed to be a rectangular strip. Here one waste pipeline (as Fig. 2 shows) must be located perpendicular to the coastal line so that its emissions have the lowest impact on marine environment. The modern plastic materials, such as polyethylene, used for the fabrication of outfall pipelines, are essentially immune to the corrosive effects of seawater and to the attack by marine organisms. Moreover, due to the Fig. 1. Teledetection by satellite. flexibility of this material, the route of an outfall pipeline can easily avoid obstacles, hazards and environmentally sensitive areas. In planning a marine outfall the first step should be to determine a suitable location for the diffuser. The determination of the location and design of the diffuser should be based on obtaining adequate distance from sensitive areas and sufficient depth, dispersion and/or die-off of pollutants commensurate with the level of treatment prior to discharge to assure negligible environmental or health impact (Reiff, 2002). The total cost of an outfall is dependent, to a great degree, on the length and diameter of the outfall pipeline. The length can be determined by the location of environmentally sensitive areas, beaches and water sport areas, the slope of the ocean floor, the prevailing wind direction and the direction and velocity of currents. In this paper, the length (xP0) of the outfall pipeline is associated to a utility function Util(x)P0 whose outline satisfies the following properties: 1. The pipeline length must be strictly positive (Util(0) = 0; Util(x)>0,"x> 0). 2. Function Util(x) is strictly increasing for x> 0 up to a threshold value x * > 0 where a maximum is attained. Within reason, the further the outfall is from the coast, the better. 3. Function Util(x) asymptotically decreases to a zero level since the excessive length puts up the cost of installation. A generalized gamma function UtilðxÞ¼xaebx;8xP0; commonly used to develop the friction factors in the models of trip distribution (Ortu ´zar and Willumsen, 2001), satisfies the previous spatial behaviour (Fig. 3). Parameters aand bare positive and have yet to be calibrated. The procedure of parameter calibration must be based on the statistical analysis of distribution of observed lengths of waste pipelines. In this sense, the Fig. 2. Marine pipe. 0246 8 10 0 0.025 0.05 0.075 0.1 0.125 0.15 Length Utility Fig. 3. An instance of the generalized gamma density function. method of least-squares can be used to determine the best fit parameters aand bfor the pipeline lengths collected in a database. Although these parameters appear non-linearly in the definition of the utility function, a transformation can be applied to obtain an appropriate linear combination in the unknown parameters. In particular, taking logarithms in the expression y=x a e bx log y¼alog xbx)log y log x¼abx log x. Therefore, given a list of statistical data (x d ,y d ), d2D, where x d indicates the length of the outfall pipeline whose utility is estimated by y d , the optimal values for the variables aand bin the least-squares sense can be calculated by means of the expressions: b¼jDjPxd log xd log yd log xd  Pxd log xd  Plog yd log xd  jDjPxd log xd  2Pxd log xd  2; a¼Plog yd log xd  Pxd log xd  2  Pxd log xd  Pxd log xd log yd log xd  jDjPxd log xd  2Pxd log xd  2. The aim of this article is to analyse the problem of locating a waste pipeline on a coastal region if forbidden regions exist in order that both the impact on marine habitats is minimized and a utility function associated to pipe length is maximized. The paper is organized as follows: Section 2describes the methodology of solving the maxmin location problem. Section 3obtains the optimum for the utility function. Section 4states the model of location in the plane in formal terms as a bi-criterion problem, showing an example. In Section 5, conclusions are presented and several extensions of this problem are introduced. 2. The maxmin location problem 2.1. Formulation Let X(x,y) be the end of a waste pipeline parallel to axis OX, inside a bounded rectangle in the first square of the Cartesian plane, whose lower left-hand vertex coincides with the origin of the coordinates. We assume for the sake of simplicity that the areas to be protected can be enclosed in the interior of circles (coral reefs) and rectangles parallel to Cartesian axes (sandbanks and meadows of Posidonia oceanica). There are two technical reasons for assuming these shapes for the sensitive regions: first, the underlying distance in the scenario is Euclidean, and subsequently the level curves of iso-affectation will be circumferences in the absence of predominant currents, and secondly, the digitization required to process the images obtained by spatial sampling in the coastal zone is based on the previous existence of a rectangular grid which supports maps of colour variations (see Caerio et al., 2003). In general coral reefs have irregular shapes (usually non-convex, close to a fractal shape) and the orientation of sandbank boundaries is dependent on the prevailing direction of the waves. Nevertheless, the possibility of overlapping those basic figures (circles and rectangles) of a small size provides a reasonable level of accuracy in the assumption. Evidently, a higher resolution in the maps will lead to a greater computational effort for the determination of solutions. When considering a medium scale, a polygonal approximation to the morphology of the protected habitats can be appropriate. Notice that only one homogeneous type of polygonal contour is assumed in the paper: forbidden regions are iso-orientated rectangles. This assumption provides an immediate decomposition of the study region into cells, which are also rectangular, where the curves containing efficient points can be regularly determined (see figure which illustrates the last case considered in the Appendix). The methodology developed for this context can be easily adapted to solve the corresponding location problem in presence of polygonal regions which are non-necessarily rectangular. Let X= [0,M]·[0,L] be a subset of R2, with M,L> 0, which denotes the region under consideration. Let C¼fCi:i2Igbe a set of forbidden circular regions, with respective centres at A i (a i1 ,a i2 ) and non-negative radii r i , which represents the set of coral reefs which need protection in relation to pollutant emissions. In addition, let R¼fRj:j2Jgbe a set of rectangular regions, with the respective lower left-hand vertices denoted by V j (v j1 ,v j2 ), length l j and height h j , which are assumed to be the Posidonia oceanica prairies or sandbanks (rectangular bands parallel to coastal fringe) considered as zones of biological interest. These circles and rectangles are completely or partially contained in X. In general, let Z¼fZk:k2Kgrepresent a set of circular or rectangular zones. The feasible region for the problem is S¼Xnð[kZkÞ. Since each Z k is a simply-connected closed set, the distance from a point Pto a zone Z k can be defined as dðP;ZkÞ¼min Q2Zk d2ðP;QÞ; where d 2 (Æ,Æ) denotes the Euclidean distance. Once the intensity of the waste dispersion decreases with the Euclidean distance from the diffuser X, the natural objective will coincide with the formulation of the corresponding 1-Maxmin problem; namely, max X2SDminðXÞmin kfdðX;ZkÞg. For a comprehensive survey of the literature on inverse optimization, the reader is referred to Qin et al. (2003). 2.2. A concise review This maxmin problem can be transformed into an equivalent formulation: max d s.t. d6dðX;ZkÞ;k2K X2S; ! max d s.t. d6min Q2Rj d2ðX;QÞ;j2J; d6min Q2Ci d2ðX;QÞ;i2I; X2S 8 > > > > > > > < > > > > > > > : Analogous formulations for the aforementioned problem can be found in the literature designed for the modelling of real scenarios. For instance, 1. Melachrinoudis and Cullinane (1985) used this maxmim objective to locate one undesirable facility within a geographical region; in particular, they applied the model to the state of Massachusetts, approximating the forbidden regions (50 existing cities) by means of circles. Karush–Kuhn–Tucker conditions were used to identify the set of local maxima. 2. Fernandez et al. (1997) used two approaches: the first methodology was based on the Karush–Kuhn– Tucker conditions and the second geometrical development exploited the concept of proximity of segment, to solve the problem: max d s.t. d6min Q2Sj d2ðX;QÞ;j2J; X2S; 8 > > > > < > > > > : where each S j was a forbidden polygonal region. The authors applied the methodology to locate an incineration plant in a region of the South of Spain where the predominant wind directions indicate a specific shape for the protected area centred at each population centre. 3. Tuy et al. (2003, in press) proposed an efficient difference-of-convex (DC) optimization algorithm to solve the maxmin problem max d s.t. d6min Q2Di d2ðX;QÞ;i2I; X2S; 8 > > > > < > > > > : where each D i was a circular region to be preserved. Two examples were presented in this last paper to illustrate the methodology followed. One of them involved 11 circles with different centres and radii, where 45 iterations were necessary so that the DC algorithm would reach the optimal solution. Fig. 4 shows the largest empty circle from the set of existing circles considered in a rectangular region. 2.3. Our methodological proposal Although different non-convex quadratic programming methods can be applied to solve this largest empty ball problem, we propose an approach based on the concept of the area Voronoi diagram (Okabe et al., 2000) to find a finite dominating set of circle centres from which global optima can be determined. The area Voronoi diagram can be constructed from its corresponding line Voronoi diagram. Let E¼ð Sk2KoZkÞ[oXrepresent the union of the set of boundaries of Zand the four edges of rectangle X. In our model the lines which delimit each forbidden zone oZ k belong to the following types: 1. Scenario boundaries; namely, axis OX (y= 0), axis OY (x= 0) and delimitations x=M,y=L. 2. Contour of circle C(A,r) centred at A(a 1 ,a 2 ) and with radius r>0. 3. Contour of rectangle R(V,l,h), whose lower left-hand vertex is located at point V(v 1 ,v 2 ) and whose horizontal and vertical dimensions are respectively l> 0 and h>0. Let joZ k jbe the number of line segments and circumference arcs which enclose protected area Z k . Define jEjas the total number of lines to be considered for building the line Voronoi diagram: jEj¼Pk2KjoZkj. The following proposition deals with the computation of the bisector curves for all possible pairs of boundary types of the aforementioned zones. Proposition 1. Bisector curves for zone pairs are piece-wise curves composed of straight lines, parabolas and/or hyperbolas. Therefore, the edges of this line Voronoi diagram are always simple continuous curves. An application of Caratheodory’s representation theorem on the plane (see Rockafellar, 1970) leads us to state that the diagram vertices may be defined as those points shared by three or more Voronoi edges. Due to this fact, the area Voronoi diagram can be efficiently generated in time OðjEjlog jEjÞ following the procedure suggested by Okabe 1 2 3 4567 8 9 10 11 LAND X Fig. 4. Largest empty circle. et al. (2000); namely, first build the line area Voronoi diagram associated to edges which represent the boundaries of forbidden zones, and then delete the superfluous edges. The aforementioned computational complexity is ensured by the property labelled OKA1 in that book (page 187). Figs. 5 and 6 respectively show the line Voronoi diagram and the area Voronoi diagram associated with the example of Tuy’s paper. Once the line Voronoi diagram has been built, the largest empty circle problem: Given a set of distinct rectangular and circular zones Zfind the largest empty circle whose centre is in set X can be solved. 3. Maximizing the utility function The two parameters aand bof function y¼UtilðxÞ¼xaebx allow the analyst to fine-tune the utility associated with the pipeline length. Its calibration can be based on the coincidence between statistical parameters of pipeline samples (for an identical purpose) and theoretical values of the utility function where relevant properties exist. Among these are: 1. Since function y=x a e bx , is a differentiable function, the study of the sign of its derivative function y0¼ðabxÞxa1ebx; in the interval x2(0,M), indicates that its only maximum is attained at point x¼a bfor a> 1. A typical length for the pipeline length average is  l¼200 m (Reiff, 2002). Therefore, the assumption a b¼ lpermits a first reduction of the number of parameters to calibrate. 1 2 3 4567 8 9 10 11 LAND Fig. 5. The line Voronoi diagram. 1 2 3 4567 8 9 10 11 LAND Fig. 6. The area Voronoi diagram. 2. Taking into account that y0¼abx xy a new derivation can be performed, yielding y00 ¼ðabxÞ2a x2y. Since y> 0, two inflexion points can easily be deduced by imposing condition y00 =0: x1¼a bffiffiffi a p b;x2¼a bþffiffiffi a p b. The second value indicates that the asymptotical decrease to zero for the utility function starts when the pipeline length exceeds a distance ffiffia p bfrom the length average  l¼a b. Together with the aforementioned property, this new property can also be used to eventually determine the parameter values. 3. Assuming an approach for a single-parameter bwhich can take value 1, hence, ais the only parameter which characterizes the maximum of the utility function. Moreover, if we are interested in a representation of the component of utility in terms of probability, a simple factor is all that is required to normalize the utility function; namely, UtilðxÞ¼ 1 Cðaþ1Þxaex;a>0. The normalization factor is based on the Euler gamma function: CðzÞ¼Z1 0 tz1etdt;z>0. The general properties of this new single-parameter version of the utility function remain the same. 4. Bi-criterion optimization In practice, bi-objective problems are often reduced to a single-objective problem by following one of two possible strategies: (a) Modelling an objective function which is a weighted sum of the individual functions. (b) Setting one objective, whose value is limited, as a constraint and then optimizing the other objective. Moreover, the bi-objective problem max DminðXÞ max UtilðXÞ s.t. X2S 8 > < > :ðDUMPÞ can also be directly tackled by constructing the set of non-dominated (efficient or Pareto-optimal) points. The approach dealt with in this paper follows this methodology. For this purpose, two useful definitions are introduced. Definition 1. A solution X0=(x0,y0) dominates the solution X=(x,y) if and only if Dmin(X0)PDmin(X), Util(X0)PUtil(X). Dominance is strict when at least one inequality is strict. Definition 2. Xis an efficient solution for the bi-objective problem if and only if no other solution X0exists which strictly dominates X. Let ESbe the set of those edges in the considered region Xwhich are outside all forbidden zones Zk(i.e., ES¼E\S) and let NSbe the node set of the zone Voronoi diagram inside the feasible region S. Proposition 2. ESis a dominating set of solutions for DUMP in S. Proof. Rectangular region Xcan be vertically scanned by means of segments of equation x=k, with 06k6M, between y= 0 and y=L. Each edge e2ESseparates Voronoi regions inside S. Let Q i (k,y i (k)) be the intersection points of vertical line x=kand ES, ordered by y-coordinates. These points constitute a finite dominating set for all points of vertical section x=kinside S: Consider two consecutive points Q i and Q i+1 which are in the contour of Voronoi region Z k (i). Let Q(k,y) be a generic point interior to Voronoi region Z k (i) inside vertical segment QiQiþ1(Fig. 7). Due to the convexity property of the Euclidean distance, the maximum of function d(Q,Z k (i)) is reached at any extreme of segment QiQiþ1. Therefore, at least one of these relation pairs holds for all points Qinterior to segment QiQiþ1 UtilðQÞ¼UtilðQiÞ; dðQ;ZkðiÞÞ <dðQi;ZkðiÞÞ; UtilðQÞ¼UtilðQiþ1Þ; dðQ;ZkðiÞÞ <dðQiþ1;ZkðiÞÞ.  Proposition is a consequence of this fact. h Proposition 2 suggests to investigate the reduction of the number of non-efficient points inside set ES. •Since each edge of the zone Voronoi diagram is either a straight line, a parabola or a hyperbola, a formal description is possible for each edge ein terms of variable y, which acts as a parameter by taking values in a certain interval. Subsequently, each edge ecan be analytically expressed as follows: x¼feðyÞ;8y2½y1ðeÞ;y2ðeÞ. •This expression permits us to analyse utility function Util(x(y)) along each edge with respect to variable y which takes values in its definition interval (i.e., "y2[y 1 (e),y 2 (e)]). The interior of the intervals where the variation of the utility is monotonous can be explored by means of the cancellation of the factors which appear when the derivative of function Util(x(y)) is calculated: d dyUtilðxðyÞÞ ¼ xa1ebxx0ðyÞðabxÞ. In order to complete the analysis, the extremes of interval [y 1 (e),y 2 (e)] are added for consideration. Let NBbe the set of intersection points between edges Eand boundaries (x=0,x=M,y= 0 and y=L) of the feasible region X. Q i+1 Q i Q 1 2 3 4567 8 9 10 Zk(i) LAND Fig. 7. Dominance along vertical segment x=k. If r 2 >r 1 then d 2 (X,A 2 )>d 2 (X,A 1 ) and hence the hyperbola branch turns around the circle with the smallest radius r 1 (see Fig. 16). 8. The bisector curve (see Fig. 17) of the pair composed of circle C(A,r) and rectangle R(V,l,h) is a continuous union of arcs of hyperbola branches (H) and parabolas arcs (P) since •The bisector curve for point V(a rectangle vertex) and the circle is expressed as d 2 (X,A)r=d 2 (X,V); it implies equation d 2 (X,A)d 2 (X,V)=rwhich identifies the branch of the hyperbola whose focal points are at points Aand V, and whose distance between vertices is r. •The bisector curve for a vertical (or horizontal) edge of rectangle R(V,l,h) and the circle is an arc of a parabola. 9. Finally, the bisector curve (see Fig. 18) corresponding to rectangle pairs R 1 (V 1 ,l 1 ,h 2 ) and R 2 (V 2 ,l 2 ,h 2 )isa continuous union of portions of mediatrix lines (L) and parabolas (P). References Calzadilla Pe ´rez, A., Damen, M.C.J., Geneletti, D., Hobma, T.W., 2002. 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