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Units of supply 20 Figure 7- Transportation network with m sources and n destinations For the formulation of this model the classical scheme proposed by Revindran et al. (1984) was adopted. Each and every source and each and every destination are represented by "nodes" and transportations roots by “arcs”. The amount of supply at source i is Ai and the amount of demand at destination j is Bj. The LP model representing the transportation problem is given by a number of equations grouped in: a) one objective function and b) a set of constraints equations. The objective function which stipulates that the sum of each cost associated to each transportation arc must be minimal is given as follows: where: Qij Cij A1 A2 Ai m n ∑∑ min( Z) = Q for i=1,2,…m; j=1,2,…n (1) i= 1 j= 1 ijC ij is the amount of water transported from source i to destination j (m 3 /yr); It is the decision variable, the incognita of the model; is the unit transportation cost between source i and destination j (Euro/m 3 /yr); m is the total number of supply sites; n the total number of demand sites; 1 2 i Am m Sources Destinations C11 ; Q11 Cmn ; Qmn Z is the value of the objective function. The minimum value is calculated trough an iterative procedure that stops when the value of Z at iteration x is larger the value of Z at iteration x-1. The "simplex" algorithm and its variations can be used for the solution of the transportation model as here formulated. 1 2 j n B1 B2 Bj Bn Units of demand
21 The objective function is subjected to a set of constrains that vary according to the specific problem. In any transportation problem they can be expressed as in equation (2), (3) and (4). For each supply site, equation (2) stipulates that the sum of the shipments from a specific source to all connected destinations cannot exceed the supply available at that specific supply site. In other words the sum of Qij transported from each source i must be smaller or equal then Ai. n ∑ j= 1 Q ij ≤ A i i =1, 2, … m (2) For each destination site, equation (3) stipulates that the sum of all the shipments to each destination must satisfy the demand at every destination, meaning that the amount Qij transported on to each destination j must be larger or equal to Bj. m ∑ i= 1 Q ≥ B , j = 1, 2, … n (3) ij j Finally, equation (4) stipulates that the decision variable cannot be negative, meaning that any transportation of "negative" amounts of good is not allowed. This avoids solutions that although would be correct from a mathematical point of view would make no sense in practical terms. Q ≥ 0, for i = 1,2,…m; j = 1,2,…n (4) ij When the total sum of supply is not equal (so that larger or smaller) than the total sum of demand the problem is unbalanced, meaning that not all demand will be met or not all available supply will be shipped. A transportation model can always be balanced using slack variables which represent fake sources and fake destination of water where the relative transportation cost is extremely high so that the algorithm will automatically discard them. Software applications automatically balance the problem using one of the many algorithms developed for this purpose. In the framework of this study, the transportation model is basically a balanced linear problem that can be solved by the regular simplex method. The application of the simplex algorithm is only one of the possible techniques to solve the model. 3.2 Uses and Function of the Transportation Model The transportation model is used to obtain a first approximation of how much water should be transported and in which directions. This is functional to the construction of a
Page 1 and 2: Ben-Gurion University of the Negev
Page 3 and 4: Abstract I The Sicilian water balan
Page 5 and 6: Tables of contents 1 INTRODUCTION 1
Page 7 and 8: Figures and tables V Figure 1 - Con
Page 9 and 10: 1 1 Introduction The Sicilian water
Page 11 and 12: 3 1.2 Water Balance Assessment One
Page 13 and 14: 5 In the initial process of any wat
Page 15 and 16: 7 almost exclusively with the use o
Page 17 and 18: Figure 2 - The study area 9 2.2 Cli
Page 19 and 20: 11 Figure 3 - Rainfall distribution
Page 21 and 22: Figure 6 - Qualitative features of
Page 23 and 24: 15 The aquifer is generally modestl
Page 25 and 26: 17 2.5.1 Potential vulnerability 5
Page 27: 3 Methodology 19 Two different mode
Page 31 and 32: 23 to the output. This is not the c
Page 33 and 34: Ri = Runoff, rivers, lakes, reservo
Page 35 and 36: U j = s ∑ r = 1 u r 27 for r = 1,
Page 37 and 38: 29 C = smoothness of the internal p
Page 39 and 40: 31 The resulting polygons of every
Page 41 and 42: 33 c) The location of supply nodes
Page 43 and 44: 35 In the network made of 13 arcs t
Page 45 and 46: 37 and 600 mm per year. Springs and
Page 47 and 48: 39 Agricultural 13.4 4.4 - - Na 17.
Page 49 and 50: Basin: TELLARO Code: 19086 Area: 38
Page 51 and 52: 43 plant: and the Fiumara Grande an
Page 53 and 54: 45 total balance (table 14) of the
Page 55 and 56: 47 Table 15 - Arc length, and eleva
Page 57 and 58: 49 4.5 The Costs of Transshipment T
Page 59 and 60: 51 Figure 16 - Transhipment optimal
Page 61 and 62: 53 general equilibrium between the
Page 63 and 64: 55 6 References Agnew C. and Anders
Page 65 and 66: 57 Commissario Delegato per l'Emerg
Page 67 and 68: 59 Peel M. C., Finlayson B. L., and
Page 69 and 70: ANNEX A Transportation script MODEL
Page 71 and 72: ANNEX C Global optimal solution fou
Page 73 and 74: 65 Global optimal solution found. T
Page 79:
71 Global optimal solution found. T
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