1st International Conference on Experiments/Process/System Modelling/Simulation/Optimization
1st IC-EpsMsO
Athens, 6-9 July, 2005
© IC- EpsMsO
LINEAR PROGRAMMING AND DYNAMIC PROGRAMMING APPLICATION IN
IRRIGATION NETWORKS DESIGN
Menelaos E. Theocharis1 , Christos D. Tzimopoulos 2 , Stauros I. Yannopoulos 2 , Maria A.
Sakellariou - Makrantonaki 3
1
Dep. οf Crop Production Tech. Educ. Inst. of Epirus, 47100 Arta, Greece, e-mail: [email protected]
2
3
Dep. of Rural and Surveying Engs, A.U.TH, 54006 Thessaloniki, Greece
Dep. οf Agric., Crop Prod., and Rural Envir., Univ. of Thessaly 38334,Volos,Greece
Keywords: Irrigation, head, network, cost, linear, dynamic
Abstract. The designating factors in the design of branched irrigation networks are the cost of pipes and the
cost of pumping. They both depend directly on the hydraulic head of the pump station. An increase of the head of
the pump involves a reduction of the construction cost of the network, as well as, an increase of the pumping
cost. It is mandatory for this reason to calculate the optimal head of the pump station, in order to derive the
minimum total cost of the irrigation network. The certain calculating methods in identified the above total cost of
a network, that have been derived are: the linear programming optimization method, the non linear
programming optimization method, the dynamic programming optimization method and the Labye’s method. All
above methods have grown independently and a comparative study between them has not yet been derived. In
this paper a comparative calculation of the optimal head of the pump station using the linear programming
method and the dynamic programming method is presented. Application and comparative evaluation in a
particular irrigation network is also developed.
1. INTRODUCTION
The problem of selecting the best arrangement for the pipe diameters and the optimal pumping head so as the
minimal total cost to be produced, has received considerable attention many years ago by the engineers who
study hydraulic works. The knowledge of the calculating procedure in order that the least cost is obtained, is a
significant factor in the design of the irrigation networks and, in general, in the management of the water
resources of a region. The classical optimization techniques, which have been proposed so long, are the
following: a) The linear programming optimization method [1,2,4,5,6,7,8], b) the nonlinear programming
optimization method [6,7,8], c) the dynamic programming method [3,6,8,9], and d) the Labye’s optimization method
[3,6,7,8]
. The common characteristic of all the above techniques is an objective function, which includes the total
cost of the network pipes, and which is optimized according to specific constraints. The decision variables that
are generally used are: the pipes diameters, the head losses, and the pipes’ lengths. As constraints are used: the
pipe lengths, and the available piezometric heads in order to cover the friction losses. In this study, a systematic
calculation procedure of the optimum pump station head is presented, using the linear programming method and
the nonlinear programming method is presented. Application and comparative evaluation in a particular
irrigation network is also developed.
3. METHODS
3.1 The linear programming optimization method
According to this method, the search for optimal solutions of hydraulic networks is carried out considering
that the pipe diameters can only be chosen in a discrete set of values corresponding to the standard ones
considered. Due to that, each pipe is divided into as many sections as there are standard diameters, the length of
these sections thus being adapted as decision variables. The least cost of the pipe network, P Ν, is obtained from
the minimal value of the objective function, meeting the specific functional and non negativity constraints.
a. The variance of the head of the pump station
The optimal head of the pump station is the one that leads to the minimum total annual cost of the project. In
the studies of irrigation networks, the determination of the optimal head of the pump station is essential, so that
the most economic project to be planned. As it is impossible to produce a continuous function between the
Menelaos E. Theocharis, Christos D. Tzimopoulos , Stauros I. Yannopoulos , Maria A. Sakellariou - Makrantonaki
optimal pump station head and the total cost of the project, the calculation of the optimal pump station head is
achieved through the following process: a) the variance of the pump station head are determined, b) using one of
the known optimisation methods, the optimal annual total cost of the project for every head of the pump station
is calculated, c) The graph ΡET.- HA is constructed and its minimum point is defined, which corresponds in the
value of Hman, This value constitutes the optimal pump station head. The variance of the values of the pump
station head is resulted as following [6]:
If in a complete route of an irrigation network the smallest allowed pipe diameters are selected, the maximum
value of the head of the pump station is resulted, in order to cover the needs of the complete route. A further
increase of this value, involves the setting of a pressure reducing valve on this route. Additionally, if in a
complete route the biggest allowed pipe diameters are selected, the minimum value of the pump station is
resulted, in order to cover the needs of the complete route. A further reduction of this value involves setting a
pump in this route. These two extreme values constitute the highest and the lowest possible value of the pump
station head respectively.
b. The objective function
The objective function is expressed by [ 2,6]:
f (X)  CX
(1)
where C is the vector giving the cost of the pipes sections in Euro per meter and X is the vector giving the
lengths of the pipes sections in meter.
The vectors C and X are determined as:
C  C1....C i ... C n 

C i  δ i1......δij ...δ ik

X  X1 ....X i ... X n T

X i  x i1......xij ... x ik
T
for i  1 , 2 ,..., n
(2)
for i  1 , 2 ,..., n and j  1 , 2 ,..., k
(3)
for i  1 , 2 ,..., n
(4)
for i  1 , 2 ,..., n and
j  1 , 2 ,..., k
(5)
where:
x11 , x12 , ... , x nk , are the decision variables in meter, ij is the cost of jth section of ith pipe in Euro per meter,
n is the total number of the pipes in the network , and k the total number of each pipe accepted diameters (= the
number of the sections in which each pipe is divided).
c. The functional constraints
The functional constraints are length constraints and friction loss constraints.
The length constraints are expressed by:
k
L i   x ij
(6)
 Δh i  H Α  h i
(7)
j1
The friction losses constraints are expressed by:
i
i 1
for all the nodes, i , where HA is the piezometric head of the water intake, hi is the minimum required
i
piezometric head at each node i. The sum  h i is taken along the length of every route i, of the network.
i 1
d. The non negativity constraints
The non negativity constraints are expressed by:
x ij  0
(8)
Menelaos E. Theocharis, Christos D. Tzimopoulos , Stauros I. Yannopoulos , Maria A. Sakellariou - Makrantonaki
e. The least cost of the pipe network
The least cost of the pipe network, P Ν, is obtained from the minimal value of the objective function. The
minimal value of the objective function, min f(X)  C X , is obtained using the simplex method. The calculation
is carried out for :
min H A  H A  Z A  H man  max H A
(9)
where min HA and max HA are as they are defined above in paragraph 3.1.a
f. The selection of the pump station optimal head
For the variance of the pump station head min H A  H A  Z A  H man  max H A , Pan. is calculated. Then, the
graph Pan.    is constructed from which the minimum value of Pan. is resulted. Finally the Ηman = ΗA - ΖΑ
corresponded to minP an. is calculated, which is the optimal head of the pump station.
e. Selecting the optimal pipe diameters
Using the simplex method the optimal pipe diameters, D Ν, corresponded to the optimal head of the pump
station are obtained.
3.2. The dynamic programming optimization method
According to this method, the search for optimal solutions of hydraulic networks is carried out considering
that the pipe diameters can only be chosen in a discrete set of values corresponding to the standard ones
considered. The least cost of the pipe network is obtained from the minimal value of the objective function,
meeting the specific functional and non negativity constraints.
a. The objective function
The objective function is expressed by [ 6,8,9]:

Fi*  min Δ ij  F(i*  1)

(10)
where Fi* is the optimal total cost of the network downstream the node i, and Δ ij is the cost of pipe i with a given
diameter j.
Τhe decision variables, Dij, are the possible values for the accepted diameters of each pipe.
b. The functional constraints
The functional constraints [ 6,8,9] are specific functional constraints and non negativity constraints.
The specific functional constraints for every complete route of the network, are expressed by:
πj
H π j   Δh k  H i
k i
(11)
where H π j is the required minimal piezometric head value at πj end, Hi is the required minimal
πj
piezometric head value at node i, and the sum  Δh k is the total friction losses from node i to the end πj of
k i
every complete route of the network.
The non negativity constraints are expressed by:
Δh i > 0
where i = 1... n, and n is the total number of the pipes in the network.
(12)
c. Calculating the minimal total cost of the network
At first the minimal acceptable piezometric head value hi for every node of the network is determined. If the
node has a water intake, then hi = Zi+25 m, otherwise hi = Zi = 4 m.
Thereinafter the technically acceptable head values at the nodes of the network are obtained from the minimal
value of the objective function[ 6,8,9]. The calculations begin from the end pipes and continue sequentially till the
head of the network, which is the edge of the last pipe. Because the decision variables can receive a limited
number of distinguishable discontinuous values, the algorithm that will be applied is the complete model of
Menelaos E. Theocharis, Christos D. Tzimopoulos , Stauros I. Yannopoulos , Maria A. Sakellariou - Makrantonaki
dynamic programming with backward movement (Backward Dynamic Programming, Full Discrete Dynamic
Program-ming – BDP, FDDP).
i). A network with pipes in sequence
The required piezometric head value at the upstream node of each end pipe is calculated [6], for every possible
acceptable commercial diameter, if the total friction losses are added to the acceptable piezometric head values
for the corresponding downstream end.
From the above computed head values, all those that have smaller value than the required minimum head
value at the upstream node, are rejected (Criterion A). The remaining head values are classified in declining order
and the corresponding costs that are calculated should be in ascending order. The solutions that do not meet this
restriction are rejected (Criterion B). The application of the B criterion is easier, if the checking begins from
the bigger costs. Each cost should be smaller than the precedent one otherwise it should be rejected. The
described procedure is being repeated for the next (upstream) pipes. Further every solution that leads to a
diameter smaller than the collateral possible diameter of the downstream pipe is also rejected (Criterion C).
ii). Branched networks[6]
From now on, every pipe whose downstream node is a junction node will be referred as a branched pipe.
Defining of the downstream node characteristics
The heads at the upstream nodes of the contributing pipes are calculated. The solutions, which lead to a head
value that is not acceptable for all the contributing pipes, are rejected (Criterion D). The remaining head values,
which are the acceptable ones for the branched node are classified in declining order.
The relevant cost of the downstream part of the network - which is the sum of the contributing branches costs - is
calculated for every possible node head value.
Defining of the upstream node characteristics
The total friction losses, corresponding to every diameter value of the branched pipe, are added to every head
value at the downstream node and so the acceptable heads at the upstream node are calculated.
The total cost of the network part including the branched pipe is calculated for all the possible heads at the
upstream node.
The acceptable characteristics of the upstream node are resulted using the same procedure with the one of the
pipes in sequence (criteria B and C).
The described procedure is continued for every next branched pipe till the head of the network where the total
number of possible head values is resulted.
d. Selecting the pump station optimal head
For every calculated value of the pump station head, H A, that is resulted from the backwards dynamic
programming procedure, a corresponding value of the network cost, Pan, is computed[ 6,8,9]. Then, the graph Pan.HA is constructed and from that the minimum value of Pan is resulted. Finally the Ηman = ΗA - ΖΑ that
corresponds to minPan is calculated and this is the optimal head of the pump station.
e. Selecting the optimal pipe diameters[ 6,8,9]
The friction losses, corresponding to every acceptable diameter of the first pipe of the network, are subtracted
from the optimal value of HA, and thus the possible head values at the end of the first pipe are calculated. The
biggest of these head values, which lies within the limits determined by the backwards procedure, is selected. The
diameter value that corresponds to this head value is the optimal diameter of the first pipe. This procedure has to
be repeated for every network pipe till the end pipes.
3.3 The total annual cost of the network
The total annual cost of the project is calculated by [6]:
Pan.  PN an.  PMan.  POan.  Ean.  0.0647767 PN  2388 .84QH man Euro
(13)
where PN an.  0.0647767 PN is the annual cost of the network materials , P M an.  265 .292 QH man is the annual
cost of the mechanical infrastructure , POan.  49.445 QΗ man
is the annual cost of the building infrastructure,
and Ε an.  2074 .11QΗ man is the annual operational cost of the pump station
4. APPLICATION
The optimal pump station head of the irrigation network, which is shown in Figure 1, is calculated. The
material of the pipes is PVC 10 atm. The required minimum piezometric head at each node, hi, is resulted by the
Menelaos E. Theocharis, Christos D. Tzimopoulos , Stauros I. Yannopoulos , Maria A. Sakellariou - Makrantonaki
elevation head at the network nodes, Zi, bearing in mind that the minimum required pressure head is 25 m at the
water consuming nodes and 4 m at the others. Figure 1 represents the real networks and provide geometric and
hydraulic details.
Figure 1. The real under solution network
4.1. The optimal cost of the network according to the linear programming method
a. The variance of the head of the pump station
The complete route presenting the minimum average gradient is A-39. The minimum and maximum friction
losses of all the pipes, which belong to this complete route, are calculated and the results are shown in table 1.
From the table is resulted that min H A  h 39   1.10 J i min L i   80 .80  2.77  83 .57 m
39
i  K1
and max H A  h 39   1.10 J i max L i   80 .80  21 .79  102 .59 m
39
i  K1
Table 1. The minimum and maximum head losses of all the pipes, which belong to the complete route presenting
the minimum average gradient
Li
Pipe
[m]
39
38
37
36
Κ36
225
240
240
125
190
Minimum losses Maximum losses
maxDi min1,1Ji minDi max1,1Ji
[mm]
[%]
[mm]
[%]
160
0.249
110
1.607
225
0.167
140
1.774
280
0.120
160
1.966
315
0.114
200
1.103
450
0.071
250
1.342
Li
pipe
[m]
K28
K19
K10
K1
A
185
120
200
50
0
Minimum losses Maximum losses
maxDi min1,1Ji minDi max1,1Ji
[mm]
[%]
[mm]
[%]
500
0.090
315
0.909
500
0.194
355
1.083
500
0.309
400
0.951
500
0.450
450
0.766
0
0
b. The optimal cost of the network
The minimization of the objective function is obtained using the simplex method. The calculation is made for
pump heads: HA = 84.00 m till 102.00 m, where ΖΑ=52m. The results are presented in Table 2.
Menelaos E. Theocharis, Christos D. Tzimopoulos , Stauros I. Yannopoulos , Maria A. Sakellariou - Makrantonaki
Table 2. The optimal cost of the network, PN , for pump heads HA = 84.00 m till 102.00 m.
HA
PN
[m]
[€]
84 377362
85 352158
86 335087
HA
PN
[m]
[€]
87 320760
88 309074
89 299360
HA
[m]
90
91
92
PN
[€]
290911
283707
277403
HA
[m]
93
94
95
PN
[€]
271466
266292
262157
HA
[m]
96
97
98
PN
[€]
258885
256757
254770
HA
[m]
99
100
101
PN
HA
PN
[€]
[m]
[€]
253397 102 251522
252340
251850
c. The optimal head of the pump station
The total annual cost of the project is calculated using the linear programming method and the eq. (13), for
pump heads: HA = ZA + Hman = 84.00 m till 102.00 m, where ΖΑ=52,00 m. The results are presented in Table 3.
After all, the corresponding graphs Pan. - ΗA (Figure 2) are constructed and the min Pan. is calculated. From the
minPan. the corresponding Ηman = ΗA - ΖΑ is resulted.
Table 3. The total annual cost of the project, Pan. , for HA = 84.00 m till 102.00 m.
HA
[m]
84
85
86
PN
[€]
43595
42562
42054
HA
[m]
87
88
89
PN
[€]
41726
41567
41535
HA
[m]
90
91
92
PN
[€]
41588
41717
41908
HA
[m]
93
94
95
PN
[€]
42122
42386
42715
HA
[m]
96
97
98
PN
[€]
43102
43563
44032
HA
[m]
99
100
101
PN
[€]
44543
45071
45640
HA
[m]
102
PN
[€]
46216
From Table 3 and Figure 2 it is concluded that the minimum cost of the network is minPΕΤ. = 41443 €,
produced for HA = 88.70 m with corresponding H man = 88.70 – 52.00 = 36.70 m
Note: The value HA=88.70 m have been resulted after the application of the above, for smaller subdivisions of H A
values which are not presented in Table 3.
e. Selecting the optimal pipe diameters
Using the simplex method the optimal pipe diameters, D Ν, corresponded to the optimal head of the pump
station are obtained. The results are presented in Table 4.
Table 4. The optimal pipe diameters, DN , according to the linear programming method
Length
[m]
Κ1
50
1
140
2
280
3
167
3α
183
4
120
5
235
6
240
7
122
7α
108
8
210
9
160
Pipe
DN
[mm]
Φ500
Φ160
Φ140
Φ125
Φ110
Φ225
Φ200
Φ200
Φ200
Φ160
Φ160
Φ125
Length
[m]
Κ10 200
10
135
11
40
11α
250
12
245
13
120
14
235
15
235
16
225
16α
5
17
210
18
145
Pipe
DN
[mm]
Φ450
Φ160
Φ160
Φ140
Φ125
Φ225
Φ200
Φ200
Φ200
Φ160
Φ160
Φ125
Length
[m]
Κ19 120
19
230
20
240
21
250
22
94
22α
156
23
120
24
230
25
123
25α
107
26
255
27
165
Pipe
DN
[mm]
Φ400
Φ200
Φ200
Φ160
Φ140
Φ125
Φ200
Φ200
Φ200
Φ160
Φ160
Φ125
Pipe
Κ28
28
28α
29
30
31
32
33
33α
34
35
Κ36
Length
[m]
185
131
54
240
235
255
125
64
161
235
225
190
DN
[mm]
Φ355
Φ225
Φ200
Φ200
Φ160
Φ140
Φ200
Φ200
Φ160
Φ160
Φ125
Φ280
Length
[m]
36
125
37
9
37α
231
38
240
39
225
40
120
41
240
42
225
43
80
43α
220
Pipe
DN
[mm]
Φ225
Φ225
Φ200
Φ200
Φ160
Φ200
Φ200
Φ160
Φ140
Φ125
4.2. The optimal cost of the network according to the dynamic programming method
a. The optimal cost of the network
The complete model of dynamic programming with backward movement (Backward Dynamic Programming,
Full Discrete Dynamic Program-ming – BDP, FDDP) is applied. The results are presented in Table 5.
Menelaos E. Theocharis, Christos D. Tzimopoulos , Stauros I. Yannopoulos , Maria A. Sakellariou - Makrantonaki
Table 5. The optimal cost of the network, PN , according to the dynamic programming method
HA
[m]
83,57
83,74
84,02
84,49
84,73
85,00
85,25
85,49
PN
[€]
405033
395595
384593
371598
364916
360399
353925
349940
HA
[m]
85,79
86,01
86,25
86,48
86,76
87,00
87,23
87,48
PN
[€]
345931
341371
338809
334756
330295
327143
324942
321236
HA
[m]
87,75
88,06
88,25
88,50
88,56
88,79
89,03
89,24
PN
[€]
319724
316387
312555
310703
309127
307246
306172
302955
HA
[m]
89,50
89,71
90,02
90,25
90,50
90,77
91,04
91,24
PN
[€]
300660
298800
297106
294676
292930
291689
290119
288481
HA
[m]
91,48
91,75
92,01
92,23
92,49
92,82
93,02
93,24
PN
[€]
286134
284619
283533
281717
279850
279005
277241
275839
HA
[m]
93,38
93,77
94,06
94,49
95,04
95,48
95,99
96,48
PN
[€]
274908
273359
271466
269746
267313
264974
261993
261913
HA
[m]
97,06
97,57
97,89
98,79
99,78
100,24
100,86
102,57
PN
[€]
261429
260170
258266
256869
256232
255319
254421
253858
c. The optimal head of the pump station
Using the eq. (13), the total annual cost of the project is calculated for pump heads, H A, the values given in
table 5. The results are presented in Table 6. After all, the corresponding graphs Pan. - ΗA (Figure 2) are
constructed and the min Pan. is calculated. From the minPan. the corresponding Ηman = ΗA - ΖΑ is resulted.
Table 6. The total annual cost of the project, Pan. , according to the dynamic programming method
HA
[m]
83,57
83,74
84,02
84,49
84,73
85,00
85,25
85,49
PN
[€]
45130
44619
44073
43519
43228
43096
42823
42709
HA
[m]
85,79
86,01
86,25
86,48
86,76
87,00
87,23
87,48
PN
[€]
42627
42465
42445
42321
42195
42139
42134
42040
HA
[m]
87,75
88,06
88,25
88,50
88,56
88,79
89,03
89,24
PN
[€]
42104
42075
41937
41968
41907
41920
41993
41911
HA
[m]
89,50
89,71
90,02
90,25
90,50
90,77
91,04
91,24
PN
[€]
41919
41922
41999
41978
42016
42095
42160
42175
HA
[m]
91,48
91,75
92,01
92,23
92,49
92,82
93,02
93,24
PN
[€]
42166
42227
42313
42327
42360
42500
42509
42550
HA
[m]
93,38
93,77
94,06
94,49
95,04
95,48
95,99
96,48
PN
[€]
42571
42706
42759
42905
43073
43187
43299
43583
HA
[m]
97,06
97,57
97,89
98,79
99,78
100,24
100,86
102,57
PN
[€]
43900
44126
44197
44640
45192
45412
45723
46712
From Table 6 and Figure 2 it is concluded that the minimum cost of the network is minPΕΤ. = 41907 €,
produced for HA = 88.56 m with corresponding H man = 88.56 – 52.00 = 36.56 m
d. Selecting the optimal pipe diameters
The optimal pipe diameters are calculated according to the procedure that has been described in paragraph
3.2.e. The results are presented in Table 7.
Table 7. The optimal pipe diameters, DN , according to the dynamic programming method
Pipe
Κ1
1
2
3
4
5
6
7
8
9
Length
[m]
50
140
280
350
120
235
240
230
210
160
DN
[mm]
Φ500
Φ160
Φ140
Φ125
Φ225
Φ200
Φ200
Φ200
Φ140
Φ140
Pipe
Κ10
10
11
12
13
14
15
16
17
18
Length
[m]
200
135
290
245
120
235
235
230
210
145
DN
[mm]
Φ450
Φ160
Φ160
Φ110
Φ225
Φ200
Φ200
Φ200
Φ160
Φ140
Pipe
Κ19
19
20
21
22
23
24
25
26
27
Length
[m]
120
230
240
250
250
120
230
230
255
165
DN
[mm]
Φ400
Φ200
Φ200
Φ160
Φ140
Φ200
Φ200
Φ200
Φ160
Φ110
Pipe
Κ28
28
29
30
31
32
33
34
35
Κ36
Length
[m]
185
185
240
235
255
125
225
235
225
190
DN
[mm]
Φ355
Φ225
Φ200
Φ160
Φ140
Φ200
Φ200
Φ160
Φ110
Φ280
Pipe
36
37
38
39
40
41
42
43
Length
[m]
125
240
240
225
120
240
225
300
DN
[mm]
Φ225
Φ225
Φ200
Φ140
Φ225
Φ200
Φ160
Φ125
Total
annual cost
cost of
Total
annual
ofthe
theproject
project
3 3
P Pan
10 Euro]
€]
an.. [ [x10
Menelaos E. Theocharis, Christos D. Tzimopoulos , Stauros I. Yannopoulos , Maria A. Sakellariou - Makrantonaki
46,50
46,00
45,50
45,00
44,50
44,00
43,50
43,00
42,50
42,00
41,50
41,00
40,50
Dynamic method
Linear method
83
84
85
86
87
88
89
90
91
92
93
94
95
96
97
98
99 100 101 102 103
Pump
station
ΗΑ [m]
Pump station
head
HAhead
[m]
Figure 2. The optimal total annual cost of the network, P an, per the pump station head, H A.
5. CONCLUSIONS
The search of the optimal head of the pump station using the linear programming method, should be realized
between the acceptable values, which are resulted from the complete route presenting the minimum average
gradient.
The selection of optimal pipe diameters using both the optimization methods, results a vast number of
complex calculations. For this reason the application as well as the supervision of these methods and the control
of the calculations is time-consuming and difficult, especially in the case of using the dynamic programming
method in a network with many branches.
The optimal head of the pump station that results, according to the linear programming method is Hman =
36.70 m, while according the dynamic programming method is Hman = 36,56 m. These two values differ only
0.38 %. while the corresponding difference in the total annual cost of the project is only 1.12 %.
The two optimization methods in fact conclude to the same result and therefore can be applied with no
distinction in the studying of the branched hydraulic networks.
6. REFERENCES
[1] Alperovits E. and Shamir U., 1977. Design of optimal water distribution. Water Res. Res. l3 (6): 885-900.
[2] Ioannidis, D. A., 1992. "Ανάλυση και εφαρμογή του γραμμικού προγραμματισμού σε συλλογικά δίκτυα υπό
πίεση και σύγκριση με τη μη γραμμική μέθοδο και τη μέθοδο του Labye", M.Sc. Thesis, A.U.TH., Salonika, (In
Greek).
[3] Labye Y., 1966, Etude des procedés de calcul ayant pour but de rendre minimal le cout d’un reseau de
distribution d’eau sous pression, La Houille Blanche,5: 577-583.
[4] Shamir U., 1974. Optimal design and operation. Water Res. Res, 10(1): 27-36.
[5] Smith D. V., 1966. Minimum cost design of linearly restrained water distribution networks. M. Sc. Thesis ,
Dept. of Civil Eng., Mass. Inst. of Techncl., Cambridge.
[6] Theocharis, M. (2004). “Βελτιστοποίηση των αρδευτικών δικτύων. Επιλογή των οικονομικών διαμέτρων”,
Ph.D. Thesis, Dep. of Rural and Surveying Engin. A.U.TH., Salonika, (In Greek).
[7] Tzimopoulos, C. (1982). “Γεωργική Υδραυλική ” Vol. II, 51-94, Salonika, (In Greek).
[8] Vamvakeridou - Liroudia L., 1990. "Δίκτυα υδρεύσεων - αρδεύσεων υπό πίεση. Επίλυση – βελτιστοποίηση",
Athens, (In Greek).
[9] Yakowitz, S., 1982. Dynamic programming applications in water resources, Journal Water Resources.
Research, Vol. 18, Νο 4, Aug. 1982, pp. 673- 696.