OPTIMIZING THE PRESSURIZED IRRIGATION NETWORKS USING FUZZY
LINEAR AND DYNAMIC PROGRAMMING
S. CHONDROGIANNIS*, C. TZIMOPOULOS* AND C. EVANGELIDES*
* Laboratory of Hydraulic Works and Environmental Management
Aristotle University of Thessaloniki-GREECE-GR- 54124
E-mail:sokchondro@yahoo.gr
EXTENDED ABSTRACT
The aim of this study was the optimization of pressurized irrigation networks using
conventional and fuzzy numbers. In particular, the method of dynamic programming was
utilized, the implementation of which required the development of a program in Visual
Fortran. The program was first applied using conventional numbers in three irrigation
networks in the area of Kavasila (the former lake Giannitsa) and the results were
compared with the results obtained by linear programming.
Fuzzy logic was then applied to the same problem, since some parameters of the
problem contain uncertainty. The problem was first approached utilizing fuzzy linear
programming and consequently a fuzzy dynamic programming methodology was
constructed for the case of irrigation networks. The parameter with uncertainty in our
problem was assumed to be the pressure at hydrants.
The conclusions were as follows: a) in case of solving irrigation networks by conventional
numbers, the method of dynamic programming gives the best results for irrigation
networks having more than 80 pipelines, b) for smaller networks, the use of linear
programming is preferable, c) in case of fuzzy numbers, the fuzzy dynamic programming
is a very efficient method, satisfying fully the uncertainties of the problem and presenting
a good flexibility with respect to the size of calculations, especially for networks having
more than 40 pipelines.
Keywords
Pressurized irrigation networks, optimization, dynamic programming, linear programming,
fuzzy dynamic programming, fuzzy linear programming.
1. INTRODUCTION
The issue of minimizing the cost of irrigation networks, has been puzzling Hydraulic
Engineers for many years. To deal with this problem, they developed the following main
methods:
a. Linear programming. According to this method there is an objective function which is to
be maximized (or minimized), subject to some constraints. Both the objective function
and constraints are linear. The unknown variables of the problem (decision variables) are
usually the lengths of pipes (Smith, 1966; Karmeli et al., 1968; Robinson & Austin, 1976;
Alperovits & Shamir, 1977; Shamir and Howard, 1977; Tzimopoulos & Ioannides 1997;
Theocharis 2004).
b. Non-linear programming. This method optimizes a nonlinear equation using Lagrange
multipliers or other methods. Usually the unknown variables of the problem are the
hydraulic pressure values at the nodes (Noutsopoulos, 1969; Swamee et al, 1973;
Tzimopoulos, 1982, Theocharis, 2004).
c. Dynamic programming. According to this method, the possible solutions to the problem
are examined step by step. The decision variables of the problem are the possible values
of available diameters the pipes can have (Liang, 1971; Yang et al., 1975; Yakowitz,
1982). The Labye method is a special case of the dynamic programming method
(graphical approach). (Labye, 1966; Livaditis, 1972; Vamvakeridou-Lyroudia, 1990;
Tzimopoulos, 1991; Theocharis, 2004).
Figure 1: Thessaloniki’s plain
The purpose of this research, is to minimize the cost of pressure irrigation networks, with
crisp and fuzzy data. A dynamic programming program for solving irrigation networks
written in Visual Fortran was designed (DYNARD) and applied to three irrigation networks
in the area of Kavasila (Figure 1). These results were compared with those obtained from
the linear programming (Chondrogiannis, 2005). After that, the fuzziness has been
introduced to the values of the required pressure at the hydrant.
2. Linear Programming and how it applies at irrigation networks
The method of linear programming is well known. The method applies at irrigation
networks in the following way. The unknown parameters of the problem refer to the
lengths of every possible diameter that can be used (satisfy the velocity requirements) for
every pipeline. The objective function which must be optimized is referring to the total
cost of the network. This cost has to be minimized:
in f  x   c1,1 x1  c1, 2 x 2  ...  c n ,m x N
There are three types of constrains:
a. Length constrains: There is one constrain of that type for every pipe of the network. If
the first pipe for example has four possible diameters than the constrain of that pipe
is: X 1  X 2  X 3  X 4  L1
b. Pressure constrains: There is one constrain of that type for every hydrant of the
network. It is necessary that the total pressure loss between the head of the network
and the current hydrant is less or equal to the greater pressure loss that still makes
that hydrant work correctly: a1,1 X 1  a1,2 X 2  ...  an ,m X N  b j  H A  H i ,req .


c. Since unknown parameters of the problem refer to lengths of the pipe with specific
diameters, all the possible solutions must be xi  0
3. DYNAMIC PROGRAMMING
3.1 General
Dynamic programming is a computational method, which can be applied when we have
to solve a problem, resulting from the composition of individual interdependent decisions.
The theory of dynamic programming has been developed by Bellman (1956, 1956a,
1957), who introduced the term “Dynamic Programming” and interpreted it as the way for
solving problems, in which we must take a series of decisions and each of them affecting
future decisions and all together creating an optimum result.
3.2 Mathematic definition and solving of the problem
Let us assume that we have a certain amount of a resource available (eg money,
manpower, materials, etc.). This quantity is distributed into various ways which are called
activities. Stands for:
 N: the number of steps
 i (x i ) : the performance function from xi decision in i step to the objective
function,
 x: the total resource amount which is available (x: state variable)
 An(xn): the cumulative value (of the objective function) until stage n, where at n
stage, we spend xn of the resource,
 fn(xn): the maximum cumulative yield up to this stage (at this stage the available
quantity xn is limited by the equation xn= x-xn because of the previous steps).
We want to find the values of xi for i = 1,2, ..., N, which maximize F:
F ( x1 , x 2 ,.....x N )   1 ( x1 )   2 ( x 2 )  .......   N ( x N ) , (→ usually not linear)
 x  x 2  .......  x N  x
with the constraints:  1
.
 xi  0  i  1,2,.....N
Usually the problems don’t resemble multistage decision-making processes. However
they can be converted easily to that format, assuming that the resource x is distributed
sequentially across different activities.
For solving a dynamic programming problem, we assume that at the first step we spend
x1 of the resource, at the second x2, at the third x3, etc. So we can match:
 Activities  steps of the problem,
 Situation at each stage  possible decisions we can make during this step,
 Decisions at each stage  quantity we can spend during this step.
Furthermore, as already mentioned, we symbolize with An(xn) the cumulative value of the
objective function at the stage n, where at the n stage, we have spent xn quantity of the
resource and with fn(xn) the maximum cumulative value of the objective function at this
stage. This has as a result the following equations:
a1 x1 
for n  1

An x n   
a n x n   f n 1 x n 1  for n  2,3,..., N
f n x n   maxAn x n 
for n  1,2,.., N ,
where 0  xn  x  xn1
for n=1,2,…,N.
4. APPLICATION OF THE METHOD TO IRRIGATION NETWORKS
The method that has been chosen for the design of the irrigation networks is the
complete method of dynamic programming with discrete values (Full Discrete Dynamic
Programming - FDDP, Howard, 1957, Blackwell, 1962, Vamvakeridou-Lyroudia, 1990).
The state variable definition is very crusial in all dynamic programming problems. In the
case of irrigation networks we select the hydraulic pressure as a state variable which is
the determinant factor for the choice of diameters (decision variables of the problem).
The implementation of this method at irrigation networks is similar to the way someone
can solve the problem of optimal path. In a problem of economic optimization of an
irrigation network, we know that in the main route, the optimal solution at its end (last
hydrant) should have operating pressure equal to the required pressure of the hydrant.
Therefore the process will start at the end (the final state is given and we want to find
optimal solutions for the initial conditions → we start from the end of the process).
Firstly for each pipe (each step) and for all available diameters the water velocity is
calculated. The diameters which lead to a velocity greater than the regulations allow are
discarded. Consequantly for the remaining solutions the following constraints are applied.
1) Discard the solutions that lead to a pressure lower than the required operating
pressure at hydrants. 2) classifying the pressure loss in descending order, then the
related costs must be distributed in ascending order. In that way, a test is carried out to
verify if higher cost ensures less pressure loss at nodes. 3) If there are branching nodes,
all the possible combinations are checked. In every combination the minimum water
pressure of all contributor pipelines is taken. The solutions that lead to water pressure
less than the maximum of the minimum are discarded. Thereby the requirements for all
pipes are ensured. 4) The solutions that lead to diameters smaller than the diameters
which have already be selected downstream are discarded.
In this way, as the process progresses from downstream to upstream, we reject some of
the possible solutions that would be considered normal, without the imposed restrictions.
This is necessary because if some solutions are not rejected, the number of possible
solutions that the problem would have to calculate would be enormous. For example, a
small network with 50 pipelines, where each pipe has five possible solutions of available
diameters, has 550  8,88178  10 34 possible solutions (of course this number of solutions
is prohibitive to check).
Figure 2: The logical diagram of DYNARD.
5. APPLICATION OF DYNAMIC PROGRAMMING TO IRRIGATION NETWORK
KAVASILA IMATHIAS
The irrigation network of Kavasila in Imathia consists of three sub-networks with a pump
station at each one of them that covers an area of approximately 350 ha. The method of
dynamic programming is solved by use of the program DYNARD, which can be applied to
any branching network. The necessary data that the program requires are: 1) the
available diameters, 2) the maximum and minimum allowed water velocities that the
regulations require for each diameter, 3) the water supply of the pipes 4) the pressure
that is desired at the beginning of the network, 5) the geometric characteristics of the
irrigation network and of the nodes - the length of the pipes located between nodes etc,
6) the necessary water pressure at network nodes and at hydrants, 7) branching nodes
and pipes that are downstream of that node. The logical diagram of DYNARD is shown in
Figure 2.
To test the method, the results from DYNARD (dynamic programming) were compared to
the results from solving the same networks using linear programming. Consequently the
program was implemented to the three subnetworks: AI, AII, AIII (Figure 3a shows AII).
The comparison of the final results of the two methods using the market cost of the
selected diameters is shown in Table 1.
AII
152
Π
151
150
149
141
O
140
139
o
148
147
136
135
124
ξ
146
137
145
144
143
142
138
122
134
133
132
131
130
129
128
Ξ
127
126
125
123
121
109
120
108
119
107
118
106
117
105
116
104
N
M
115
103
114
113
102
101
ν
μ
112
100
111
99
110
98
A
20
17
11
4
18
12
5
19
90
97
96
95
94
93
92
Λ
λ
89
91
85
a.
84
83
82
81
80
79
K
78
77
88
87
86
16
6
15
14
13
7
6
10
3
9
2
8
b.
Figure 3. a)Network AII b)Network used for the fuzzy programming
1
TABLE 1: Comparison the result between dynamic programming-linear programming
Dynamic
Linear
Absolute
Irrigation
Programming Programming
relative
network
(cost €)
(cost €)
error
Αια
302 256.8
302 788.0
ΑΙ
0.001796
Αιβ
337 318.7
337 938.3
ΑΙΙ
ΑΙΙ
716 426.4
716 365.6
0.000085
ΑΙΙΙα
671 986.9
671 447.2
ΑΙΙΙ
0.000725
ΑΙΙΙβ
72 811.5
72 811.5
6. FUZZY LOGIC
The developer of the fuzzy sets theory is Lotfi Zadeh (1965). In contrast to the Aristotelian
logic (two-valued logic), according to which sets have two possibilities (a logical sentence
can take only two values, either it can be true or false, 1 or 0), the fuzzy logic (multivalued
logic) has a value fluctuating between 0 and 1, so they are infinite values in [0,1]. The
theory of Fuzzy Logic can be applied to problems which have some degree of uncertainty
concerning their data. In the case -for example- of an irrigation network, there is
uncertainty concerning the required water pressure of the hydrants. This uncertainty may
be due either to the change of the desired pressure at the hydrants (with the maximum
pressure raised accordingly), or even change in water pressure during the operation
phase of the network. For instance the change of water level in tank, which causes a
lower water pressure at the hydrants, can be compenseted by raising the water pressure
requirements of the hydrants during the design phase of the network (with maximum
pressure raised at hydrants in proportion to the tank level drops).
6.1 Fuzzy Linear Programming
According to Zimmerman at a classical linear programming problem, if we input fuzziness
the problem is converted to a goal problem of the following type:
Max λ
λ  μ i B  x , i  (1,2,..., n  1) ,
x0

 c 
where B    and μi is called membership function and is equal to (pi: maximum
 A
tolerance of constrain i) :

( B  x ) i  d i  pi , greater tolerance
0
 

 i ( Bx )  [0,1] d i  ( B  x ) i  d i  pi

1
(B  x)i  di
i  1,2,......, n  1

At irrigation network problems the objective function is crisp and there is a fuzziness at
constrains). According to Zimmerman this is a three steps solution. The first two steps are
used to calculate the maximum tolerance zo of the objective function so that the
membership function of it can be found. Firstly, the problem without fuzziness is
calculated. After that the problem with the maximum tolerances is calculated. Finally, the
fuzzy linear programming problem at step 3 is calculated.
6.2. Fuzzy dynamic programming in irrigation network
As it can be seen, dynamic programming is not a standard procedure. In the same way,
the Fuzzy Dynamic Programming has not a standard theory. The logic of the method is
the following. The dynamic programming process is used to solve the network. Firstly the
problem is solved without tolerances. So the cheapest solution is found which is obtained
when we have Hreq at hydrants. After that, the problem is solved having the maxima
tolerances that can be accepted (That means that less pressure losses exist in the
network, and so higher pressure at hydrants is obtained). After solving the network, all
the possible solutions have to be considered. The cheapest possible solution gives
pressure at its beginning equal to the necessary pressure (Hreq). In all other solutions the
cost gradually rises and at the same time the pressure losses in the network fall. For
every different pressure value at the beginning of the network a different pressure at
hydrants exists. For every hydrant of the network a membership function
i  1
H i  H req
pi
can be calculated between the values Hreq and Hreq+tolerance. So, every solution has its
λ=min(μi), which is the lowest μi calculated from all the hydrants. The lowest μi is selected
because this hydrant uses the greatest percentage of tolerance which characterizes the
network as well. So, every possible solution has its own λ and we pick the highest
λ΄=maxmin(μi) from all the solutions, which verifies all the fuzzy constraints. This is the
solution of the fuzzy dynamic programming.
6.3 Application of fuzzy linear and fuzzy dynamic programming
An irrigation network of 20 pipes was selected (Figure 3b), where the numbers indicate
the number of pipes. (For the example pipe PVC 10atm were used with kinematic
viscosity ν=1.16*10-6 m2/s and roughness factor Κ=7.5*10-5m).
Solution with fuzzy linear programming
Step 1: f = f1 = 116376.10 € (Problem without fuzziness)
Step 2: f = f0 =124119.00 € (Problem after the introduction of tolerances)
Step 3: 119814.30 € (for θ=0.556→λ=0.444) - Final fuzzy solution.
Solution with fuzzy dynamic programming (with the use of the program DYNARD)
Step 1: 117 682.50 € (Problem without fuzziness)
Step 2: 124 779.70 € (Problem after the introduction of tolerances)
Step 3: 121758.80 € (for λ = 0.480) - Final fuzzy solution.
The reason why linear programming gives cheaper solutions is that the network is very
small, so the fact that dynamic programming changes the values of diameters only
between nodes in contrast to the method of linear programming that can split a pipe into
two secondary pipes with corresponding diameters is a crisp parameter.
7. CONCLUSIONS
The following conclusions are drawn:
1) In case of solving irrigation networks by conventional numbers, the method of dynamic
programming gives the best results for irrigation networks having more than 80 pipelines.
For small irrigation networks (about 30-40 pipelines or less) the method of linear
programming gives better results because at small networks the fact that dynamic
programming doesn’t choose two diameters for any pipeline leads to more expensive
solutions. For medium irrigation networks dynamic programming is preferable. The
reasons are: a) It can solve a problem of irrigation networks in few minutes, which almost
coincides with the optimal solution obtained using linear programming. b) It does not
require any special knowledge of hydraulics. The only data needed is the correct input
data of the problem c) In cases of simple irrigation networks it is much faster and there
isn’t danger for making mistakes. (It should be emphasized that for a problem with 80
pipes (Figure 3a), writing the objective function in the case of Linear Programming
requires several pages, with the omnipresent risk of making an error).
2) The introduction of Fuzzy numbers in the same network problem on the other hand
has many advantages because the problem has some imprecise data. The results
obtained under uncertainty based on tolerances, are closer to the real life problem. Fuzzy
dynamic programming is a very efficient method, satisfying fully the uncertainties of the
problem and presenting a good flexibility with respect to the size of calculations,
especially for networks having more than 40 pipelines. This happens because it is very
difficult to solve a goal problem if its equations are very big and even if someone could
manage to solve it, the model is very unstable (it can find topical minimum). On the other
hand, fuzzy dynamic programming can even handle big irrigation networks without any
problem. For small networks, fuzzy linear programming leading to better results as was
seeing in the example.
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