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A single-period model for conjunctive use of ground and surface water under severe overdrafts and water deficit
European Journal of Operational Research 133 (2001) 653±666
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Theory and Methodology
A single-period model for conjunctive use of ground and surface water under severe overdrafts and water de®cit M.N. Azaiez
*,1,
M. Hariga
Industrial Engineering Program, King Saud University, P.O. Box 800, Riyadh 11421, Saudi Arabia Received 19 July 1999; accepted 7 July 2000
Abstract We develop a model for a multi-reservoir system, where the in¯ow to the main reservoir and the demand for irrigation water at local areas are stochastic. The controlled releases from the main reservoir to local ones can be supplemented by groundwater from an aquifer suering from severe overdrafts. We consider high penalty costs for pumping groundwater in order to reduce the risk of total depletion of the aquifer as well as quality degradation and seawater intrusion. We analyze the problem for a single period with a single decision-maker approach. We allow de®cit irrigation in the problem of maximizing expected total pro®t for the entire region. We show that the formulated nonlinear stochastic problem is concave with linear constraints and propose an iterative procedure that generates an optimal operating policy. We illustrate the model through a hypothetical example. Ó 2001 Elsevier Science B.V. All rights reserved. Keywords: Reservoir management; De®cit irrigation; Stochastic supply; Stochastic demand
1. Introduction The region of interest, Mornag located at 20 km south of Tunis, initially imports water from the north of Tunisia to supplement for the shortage of groundwater, essentially the only source of irrigation water in the region. With the ever-increasing demand for irrigation water, overdrafts of * Corresponding author. Tel.: +966-1-4676828; fax: +966-14676652. E-mail address: [email protected] (M.N. Azaiez). 1 Currently on leave from Ecole Superieure de Commerce de Sfax, Tunisia.
groundwater have been observed during the last decades leading to serious threats of severe quality degradation and in some places to seawater intrusion, in addition to the risk of total depletion of the aquifer. To put some restrictions on groundwater withdrawals, the authorities established ``banning zones'' where it is disallowed to withdraw groundwater. At the same time, more water was imported from the north through the ``Medjerda-CapBon'' canal. Recently, several regions have been connected to this canal in order to reduce their respective local water de®cit. As a result, the canal became unable to totally ful®ll all the demands. Therefore, the supply of water from the
0377-2217/01/$ - see front matter Ó 2001 Elsevier Science B.V. All rights reserved. PII: S 0 3 7 7 - 2 2 1 7 ( 0 0 ) 0 0 2 1 2 - 5
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north has become stochastic according to the availability of water and to the aggregate demand of all regions connected to the canal. However, the local authorities in the region of interest plan to make the imported water as the main source of water supply and the groundwater as a supplemental source that can be used only when it is absolutely necessary. This plan is expected to occur at least for several seasons, until the ground stock is suciently replenished from natural and arti®cial recharges. This can be achieved by signi®cantly increasing the fees of withdrawing groundwater. The imported water to the region of interest is stored in a main reservoir. Then, releases are made to dierent local reservoirs to irrigate the local areas. The problem is to determine the optimal release policy from the main reservoir, possibly with supplemented groundwater, to local reservoirs and from local reservoirs to irrigated areas so that the total pro®t of the region is maximized. A stochastic supply of water from the north and a stochastic demand of water in each irrigated area characterize this special problem of conjunctive use of ground and (imported) surface water. The demand of irrigation water depends on prior rainfall and cropping decisions for the season or irrigation period. Such a period is usually of 8±10 weeks. Water al-
location from the main reservoir to the local ones is made at the beginning of each irrigation period. Therefore, water demand at local reservoirs is stochastic at the decision phase of water allocation from the main reservoir to the local ones. However, this demand is known at the irrigation phase. The uncertainty about the demand increases in the case of a multi-period allocation. In the present study, we are interested to analyze this problem from a single decision-maker approach (local authorities). We will restrict attention to a single season. The situation is depicted in Fig. 1. 2. Literature review There is a large number of published studies in reservoir management and operations models. These include Dudley (1988a,b), Wang and Adams (1986), Adig uzel and Cokunoglu (1984), Vedula and Mujumdar (1992), and Vedula and Nagesh Kumar (1996) to name only a few. Yeh (1985) provides a state-of-the-art review of reservoir management models. With the exception of few models, such as the one by Eiger and Shamir (1991), most of reservoir management models consider only the supply or the demand as stochastic. This is due to the added
Fig. 1. A description of the problem.
M.N. Azaiez, M. Hariga / European Journal of Operational Research 133 (2001) 653±666
complexities by treating both demand and supply as stochastic. Also, as for the present study, most of the models developed for reservoir operations for irrigation consider a single decision-maker approach (e.g., Dudley, 1988a; Vedula and Mujumdar, 1992; Vedula and Nagesh Kumar, 1996; Haouari and Azaiez, 2001). Dudley (1988b), however, develops a model for irrigation reservoir and farm management with a multi-decision-maker approach, namely a reservoir manager and individual irrigation farmers. A lot of eort has been devoted to groundwater management (for instance, Aguado et al., 1974; Aguado and Remson, 1980; Willis and Newman, 1977; Remson and Gorelick, 1980; Willis and Liu, 1984; Wanakule and Mays, 1986). Many of the groundwater management models use simulation combined with optimization, resulting in the socalled simulation±management models (Wanakule and Mays, 1986). Gorelick (1983) reviews the literature for such models. De®cit irrigation is gaining interest in irrigation planning. A number of models have been developed in the literature, where crops are deliberately under irrigated in order to save water and increase the irrigated area and possibly the pro®t. Stewart and Hagan (1973) develop a multiplicative formula for crop yield as a function of applied irrigation water. This formula is widely used in de®cit irrigation. Doorenbos and Kassam (1979) analyze experimental data on crop yield response to water, and empirically derive yield response factors Ki . These factors are sensitivity indices for water stress in the speci®ed growth stage i of the crop. Rao et al. (1988) compare three dated water-production function models and conclude that a simple heuristic multiplicative form is applicable over a wide range of stress conditions. Vedula and Mujumdar (1992) and Mannocchi and Mecarelli (1994) develop procedures to assess crop yields as a function of applied water, in terms of the ratio of actual to maximum evapotranspiration. 3. The model The main reservoir receives a stochastic quantity of water from the north. The decision-
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maker pays a price proportional to the quantity received, and allocates an amount of water to each local reservoir. If the total amount of water to be allocated to these local reservoirs exceeds the active capacity of the main reservoir, then an emergency withdrawal from the aquifer is applied to supplement for the shortage. Otherwise, the amount of water remaining in the main reservoir will be maintained for future uses. The decisionmaker will also maintain surplus water at each local reservoir if the obtained total stock of water exceeds the local demand. This action, however, would be unlikely to happen due to the constant de®cit of irrigation water in the region. If a shortage occurs, then this would be re¯ected in the response to yield for the period of interest. We will neglect the delivery costs from the main reservoir to the local ones, as these costs are minor. The problem is to identify the optimal operating policy that permits the maximization of the total contribution to the pro®t during the period of interest. An operating policy consists of three decision stages: 1. Identi®cation of the appropriate amount of water to be allocated to the entire region. 2. Allocation of this water optimally among the dierent local reservoirs for the period of interest. 3. Allocation of water at local reservoirs optimally among the selected crops. In this paper, we are interested to analyze the problem for a single period (or irrigation season). Recently, Azaiez et al. (2001) extend the problem to a planning horizon of several periods. Other possible extensions would include a multi-decision-maker approach, involving the authorities and the farmers. Our single-period problem bears some similarities with the stochastic two-echelon, singleperiod inventory problem (see Silver et al., 1998), where the main reservoir represents the central warehouse and local reservoirs represent the retailers. However, our problem has an added feature consisting of a stochastic supply to the main reservoir (the central warehouse in the inventory problem). In this analogy between the stochastic two-echelon, single-period inventory problem and
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the current problem of conjunctive use of ground and surface water, the holding costs have two components, direct and opportunity costs. Direct costs are represented by the costs of maintaining water in the dierent reservoirs for future uses (chlorinating and inspection). Opportunity costs consist of over-allocating water at the expense of reducing other users' activities. These opportunity costs are very important if the over-allocation has totally or partially been supplied from an emergency withdrawal of groundwater. To reduce the computational burden, we will assume that such holding costs are linear. Also, shortage cost consists of the important fee of an emergency withdrawal from the aquifer. Another indirect shortage cost, incurred at local reservoirs, is re¯ected in the reduced yield of the crops to be grown as a result of de®cit irrigation. This can be considered as an opportunity cost. No salvage cost is assumed. 3.1. Notation and assumptions The in¯ow of water to the main reservoir for the period of interest is stochastic. Therefore, we set the following: w: the stochastic in¯ow to the main reservoir, fw : the probability distribution function (p.d.f.) of w, Fw : the cumulative distribution function (c.d.f.) of w, lw : the expected value of the random variable w. The demand for water of each selected crop at a local reservoir is also stochastic. Therefore, we also set for each local reservoir i 1 6 i 6 n, where n is the number of local reservoirs), and for each crop j (1 6 j 6 ki , where ki is the number of selected crops to be grown on local area i): Di;j : the stochastic demand for water of crop j per unit of area at local area i for the period of interest, fi;j : the p.d.f. of Di;j , Fi;j : the c.d.f. of Di;j . Since the decision-maker will decide on the total water to be allocated (release from main reservoir combined possibly with groundwater), on the share of each reservoir i, as well as on the ir-
rigation level to be assigned to each selected crop, we let the decision variables be: R: total water to be released from the main reservoir, Ri : water to be released to reservoir i (share of reservoir i) for 1 6 i 6 n, IRi;j : irrigation level per unit of area to be allocated to crop j at local area i for 1 6 j 6 ki ; 1 6 i 6 n. The cost parameters are given by: C0 : unit cost of water imported from the north, p: unit cost of an emergency order (penalty for withdrawing groundwater, expressing the shortage cost), h: unit holding cost at the main reservoir (cost of maintaining water for future use), hi : unit holding cost of maintaining reservoir i for 1 6 i 6 n. Moreover, we let I: initial amount of water in the main reservoir before receiving the stochastic quantity of water from the north, Ii : initial amount of water in the ith reservoir before receiving its share from the main reservoir, Yi;j IRi;j ; Di;j : the expected yield of crop j per unit area at reservoir i, given some demand per unit area Di;j , during the period of interest, Ai;j : area allocated to crop j in local area i, Pi;j : pro®t per unit area of crop j grown at local area i. We assume that Yi;j IRi;j ; Di;j is concave and twice dierentiable. This assumption is not restrictive given the typical shape of Yi;j IRi;j ; Di;j as a function of applied irrigation water. This shape is given in Fig. 3. We also assume that area Ai;j is allocated to crop j independently of the amount of water to be made available. This can be justi®ed, for instance, from the nature of the soil, rotation constraints and input factors. Thus, the decision variables must satisfy n X
Ri R:
1
i1 ki X j1
Ai;j IRi;j Ii Ri
81 6 i 6 n:
2
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3.2. The cost function
3.3. The contribution to the yield
We will consider the following components of the expected total cost for the period of interest: 1. expected cost of imported water is equal to
The contribution to the yield during the period of interest of a given selected crop j at the ith reservoir will depend on the irrigation level to be allocated to this crop. This irrigation level would depend on the demand for water of the crop, and would be constrained by the initial stock, Ii , and the share, Ri , as given in (2). It is reasonable to consider the contribution to the yield as a predetermined fraction of the total yield. This can be justi®ed from Stewart's multiplicative formula (see for example, Mannocchi and Mecarelli, 1994). Alternatively, we may consider that the period of interest is the only dry period, in which shortage of irrigation water may occur. Consequently, the period of interest would be responsible for any reduction in yield that may occur. We would adopt this latter assumption for the present study. Generally speaking, the relationship between crop yield and applied irrigation water follows a trend similar to the one provided in Fig. 2. Similar relationships between crop yield and applied water can be found in English and Nuss (1982), English (1990), and Hargraves and Samani (1984). The ®gure points out two zones corresponding, respectively, to under and over irrigation of the crop. In the ®rst zone, corresponding to de®cit type of irrigation, the relationship between applied water and crop yield is approximately linear. The second zone, characterized by the beginning of the greatest curvature in the function, indicates a diminution in the rate of increase of the crop yield due to the negative eects of soil water surplus. The contribution to the yield or the yield function for the present study will be similar to the one above in case of de®cit irrigation (®rst zone). However, if the amount of water assigned to a
C0 lw ; 2. expected holding costs at the main reservoir is equal to Z h
1 R I
w I
Rfw w dw;
3. expected shortage cost at the main reservoir is equal to Z p
R I 0
R
I
wfw w dw;
4. expected holding costs at the ith reservoir is equal to hi
ki X
Z Ai;j
IRi;j 0
j1
IRi;j
Di;j fi;j Di;j dDi;j :
In fact, as we will explain below, the operating policy will not allow a crop to be over irrigated. Therefore, the surplus of allocated water to each crop (if any) will remain in the local reservoir for future use. From the individual costs above, the expected total cost will be ETC R; IRi;j ; 1 6 i 6 n; 1 6 j 6 ki Z 1 C0 lw h w I Rfw w dw Z p Z
0
R I
R I
0 IRi;j
R
I
wfw w dw
n X i1
IRi;j
Di;j fi;j Di;j dDi;j :
hi
ki X
Ai;j
j1
3
It can be easily shown that the total expected cost ETC R; IRi;j ; 1 6 i 6 n; 1 6 j 6 ki is a convex function. Note that the shortage cost of not meeting the demand at the ith reservoir will be re¯ected later in the contribution to the yield.
Fig. 2. Crop yield as a function of applied water.
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given crop exceeds the demand, then the surplus of water will remain in the reservoir. Therefore, the yield function per unit of area in the second zone will be of the form Yi;j IRi;j ; Di;j
Yi;jmax
if Di;j 6 IRi;j :
4
In other words, the function Yi;j IRi;j ; Di;j will be increasing if IRi;j < Di;j , as in the ®rst zone of Fig. 2 and will be constant in IRi;j if IRi;j P Di;j . Thus, the shape of the function Yi;j IRi;j ; Di;j will be similar to the one given in Fig. 3. For a given crop j at local area i, the expected return generated by some assigned irrigation level IRi;j is given by Z 1 Pi;j Ai;j Yi;j IRi;j ; Di;j fi;j Di;j dDi;j ER IRi;j 0 ( Pi;j Ai;j Fi;j IRi;j Yi;jmax Z
1 IRi;j
) Yi;j IRi;j ; Di;j fi;j Di;j dDi;j : 5
Proposition 3.1. ER IRi;j is concave. Proof.
( dER IRi;j Pi;j Ai;j fi;j IRi;j Yi;jmax dIRi;j Z 1 dYi;j IRi;j ; Di;j fi;j Di;j dDi;j dIR i;j IRi;j ) Yi;j IRi;j ; IRi;j fi;j IRi;j :
6
However, Yi;j IRi;j ; IRi;j Yi;jmax , therefore, dER IRi;j dIRi;j
Z
Pi;j Ai;j
1
dYi;j IRi;j ; Di;j fi;j Di;j dDi;j : dIRi;j
IRi;j
7 It follows that d2 ER IRi;j dIR2i;j (Z
d2 Yi;j IRi;j ; Di;j fi;j Di;j dDi;j 2 IRi;j dIRi;j ) dYi;j IRi;j ; IRi;j fi;j IRi;j : dIRi;j
Pi;j Ai;j
1
Note that the second term in the right-hand side vanishes as the function Yi;j ; IRi;j is constantly equal to Yi;jmax . It is clear therefore that the ®nal expression of the second derivative of ER is nonpositive from the concavity of Yi;j and the nonnegativity of fi;j . The global optimization program of total water release from main reservoir and allocation among local reservoirs and then among selected crops is given by max
R;Ri ;IRi;j
ki n X X i1
ER IRi;j
ETC R; IRi;j ;
j1
1 6 i 6 n; 1 6 j 6 ki s:t:
ki X
Ai;j IRi;j
Ri I i ;
1 6 i 6 n;
j1 n X
Ri
R 0;
i1
R; Ri ; IRi;j P 0;
1 6 i 6 n; 1 6 j 6 ki : 8
Fig. 3. The function Yi;j IRi;j ; Di;j .
It follows from the concavity of the expected return functions, ER IRi;j , and the convexity of the expected total cost function, ETC R; IRi;j ; 1 6 i 6 n; 1 6 j 6 ki that this global program is a concave program with linear constraints. Therefore, Karush±Kuhn±Tucker (KKT) conditions are
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necessary and sucient for optimality. To reduce the computational eort in solving for the KKT conditions, we replace the dependent constraints ki X
Ai;j IRi;j
Ri I i ;
Ri
i1
R0
i1
j1
by the single constraint n X
ki X
i1
j1
Ai;j IRi;j
R
n X
g k
Ii :
Therefore, the KKT conditions will be given by h
"
hi Fi;j IRi;j
k60
for 1 6 i 6 n; 1 6 j 6 ki ; # ki n X n X X Ai;j IRi;j Ii ; R j1
i1
R h
"
IRi;j
i1
p hFw R I k 0; Z 1 oYi;j IRi;j ; Di;j Pi;j oIRi;j IRi;j
fi;j Di;j dDi;j
hi Fi;j IRi;j k 0
for 1 6 i 6 n; 1 6 j 6 ki ; R P 0;
IRi;j P 0 for 1 6 i 6 n; 1 6 j 6 ki ; 9
where k is the Lagrange-multiplier associated with the single dependent constraint. After ignoring the non-negativity constraints of the optimization problem given by (8), the KKT conditions reduce to Fw R Z Pi;j
I 1 IRi;j
h k ; hp
10
oYi;j IRi;j ; Di;j fi;j Di;j dDi;j oIRi;j
hi Fi;j IRi;j k 0;
R
n X
Ii :
12
i1
ki n X X i1
i1
p hFw R I k 6 0; Z 1 oYi;j IRi;j ; Di;j Pi;j fi;j Di;j dDi;j oIRi;j IRi;j
Ai;j IRi;j
From (10), it is easy to show that a feasible k must belong to the interval p; h. Moreover, assuming that (10) and (11) can be solved for R and IRi;j , respectively, as a function of k, then it is possible to show that the following one-dimensional function:
1 6 i 6 n;
j1 n X
ki n X X
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1 6 i 6 n; 1 6 j 6 ki ; 11
j1
Ai;j IRi;j k
R k
n X
Ii
i1
is strictly increasing. Based on these results, we provide the following line-search algorithm to solve (10)±(12). Algorithm 3.1. Step 0. Select some small real number e > 0, and set L p and U h. Step 1. Set k L U =2. Step 2. Use (10) and (11) to ®nd R k; and IRi;j k. Step 3. If jg kj < e, then stop. Else if g k > 0, then set U k and go to step 1. Else g k < 0, then set L k and go to step 1. End. We will repeatedly use Algorithm 3.1 to derive optimal operating policies in the example below. In addition, we provide a detailed illustration on the use of Algorithm 3.1 in Appendix A. In the illustration below, Algorithm 3.1 is applied to a small problem. However, our computational experience with this algorithm indicates that it is ecient for large-scale problems.
4. Illustration We will provide a simple example to show the applicability of the model. This example is mostly based on hypothetical data. This is due to the fact that real data collection is still at its very ®rst stage by now. However, we do not expect that the data
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Table 1 Distribution of selected crops in the three local areas Crop Sorghum Wheat Saower Total
Area (ha) Area 1
Area 2
Area 3
600 400 ± 1000
300 ± 700 1000
± 500 500 1000
to be used would be signi®cantly dierent from the true situation. We consider three local areas n 3, each having a total land of 1000 hectares (ha). We assume that the selected crops to be grown in these areas are given by Table 1.
4.1. The demand and supply distributions The families of normal and gamma distributions have found frequent use in water resource planning (Loucks et al., 1981). The normal distribution, as a symmetric distribution, does not always satisfactorily model phenomena such as stream¯ows, which are necessarily positive and have skewed distributions. The lognormal distribution usually ®ts better for stream¯ows. However, within a same period (for instance a wet period) the normal distribution may reasonably ®t for stream¯ows, demand for irrigation water and rainfalls. The gamma distribution, which arises naturally in many problems in hydrology, has long been used to model daily, monthly and annual stream¯ows. It also has a very reasonable shape to represent rainfall. The amount of water to be imported from the north to the region of interest depends on rainfalls in the north as well as on the demand for water in
the other regions that are connected to the canal. We will assume that the stochastic in¯ow, w, to the main reservoir for the speci®c period of interest can adequately be represented by a normal distribution. We will also use the gamma distribution to represent the demand for irrigation water in the three local areas considered in this illustration. The details are provided in Table 2. In Table 2, the parameters given for the normal distribution are the mean and the variance, and for the gamma distribution the shape and scale parameters, respectively. Note that we have considered the demand of a particular crop as independent of the local area in which the crop will be grown. 4.2. The costs We will arbitrarily assign the holding costs (for maintaining water) to be 25 Tunisian dinars (TD) in the main reservoir and 30 TD in each of the local reservoirs per thousand of cubic meters for the period of interest. The purchasing cost is 90 TD per thousand of cubic meters. We will analyze the problem for a variety of shortage costs ranging from 100 to 1000 TD per thousand of cubic meters. 4.3. The returns We set the returns in TD per tonne to be, respectively, 150, 200, and 300 for sorghum, wheat, and saower. 4.4. The yield functions If a particular crop has s growth stages, for some integer s (usually s is taken to be between 3
Table 2 Demand and supply distributions Random variable
Distribution
Mean
w D1 (Sorghum per ha) D2 (Wheat per ha) D3 (Saower per ha)
Normal 2:7; 0:72 Gamma (2.0; 2.5) Gamma (2.0; 2.0) Gamma (2.0; 1.5)
2.70 0.80 1.00 1.33
(millions of m3 ) (103 m3 /ha) (103 m3 /ha) (103 m3 /ha)
Variance 0.49 0.32 0.50 0.89
M.N. Azaiez, M. Hariga / European Journal of Operational Research 133 (2001) 653±666
and 5), then the fraction of the actual yield Ya to the maximum yield Ym , due to de®cit irrigation, can be approximated by (Mannocchi and Mecarelli, 1994) Ya Ym
s Y
1
i1
Kyi 1
ETa ETm
;
13
where i is a generic growth stage, Kyi is the yield response factor of the crop at growth stage i, ETa and ETm are, respectively, the actual and maximum evapotranspiration. In the present study, since we are only analyzing the one-growth stage case, we assume that de®cit irrigation would only occur for the particular period of interest (say the dry period). In this case, no reduction in yield would occur as a result of de®cit irrigation for the remaining growth stages and the yield function would be of the form Ya ETa 1 Ky 1 : 14 Ym ETm The fraction of the actual to the maximum evapotranspiration depends on the irrigation level to be applied compared with the irrigation demand of the crop. This dependence is approximately linear, as was illustrated in Figs. 2 and 3. If no irrigation water is applied, then ETa will result from rainfall only and the fraction of ETa =ETm will have some value d (with 0 < d < 1) that can usually be estimated empirically. For the purpose of this illustration, we will assign values for d arbitrarily (but re¯ecting the sensitivity of each crop to the shortage of arti®cial irrigation). If, however, enough water is applied to satisfy full water requirement of the crop, then ETa will simply
661
coincide with ETm . Therefore, the fraction ETa =ETm will be approximated by ETa 1 dIR d: D ETm
15
Therefore, the yield function will be approximated by Ya Ym 1
Ky 1
d 1
IR D
:
16
We assume that the period of interest is the ¯owering stage. Consequently, we are using the corresponding yield response factors. The yield function and its dierent parameters are summarized in Table 3. In Table 3, wh stands for wheat, sg for sorghum and sf for saower. Note that there is no need to specify the local area in which each crop will be grown as we assumed that this would not aect the expected yield and returns. The optimal operating policy as well as the corresponding expected pro®t are given in Table 4 for a range of shortage costs. In Table 4, the shortage cost, p, as well as all other costs, pro®ts and returns are in TD. Also, R (the total release) is in thousands of cubic meters, and IRsg ; IRwh , and IRsf (the irrigation levels per unit of area allocated to sorghum, wheat, and saower, respectively) are in thousands of cubic meters per hectare. In addition, ETR is the expected total return for the entire region, MR cost is the expected cost at main reservoir (expected holding plus shortage costs), LR costs are the aggregate expected holding costs at the local reservoirs, and Pro®t is the expected total pro®t of the region ETR ETC.
Table 3 Yields and yield response factors of the selected crops Parameters
Crop Sorghum
Wheat
Saower
Ky d Ym (t/ha) Ya (t/ha)
0.55 0.4 16 5.28 IRsg =Dsg +10.72
0.65 0.3 7 3.185 IRwh =Dwh +3.815
0.55 0.1 12 5.94 IRsf =Dsf +6.06
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Table 4 Optimal operating policy for a range of shortage costs p
R
IRsg
IRwh
IRsf
ETR
MR cost
LR cost
Pro®t
100 200 300 400 500 600 700 800 900 1000
4523.460 3708.291 3324.697 3099.604 2946.232 2832.069 2742.165 2668.568 2606.533 2553.107
1.113 0.911 0.816 0.760 0.722 0.693 0.671 0.653 0.637 0.624
1.182 0.926 0.806 0.736 0.688 0.652 0.624 0.601 0.582 0.565
2.048 1.712 1.554 1.461 1.398 1.351 1.314 1.283 1.258 1.236
7304183.50 7089560.00 6957981.00 6869941.00 6804943.00 6753782.00 6711764.00 6676199.50 6645383.00 6618222.50
182465.00 206943.50 210581.47 212438.08 214065.27 215602.52 217060.77 218441.36 219720.92 220906.20
44567.00 29956.50 23869.50 20572.50 18450.50 16940.50 15794.00 14884.50 14137.50 13509.50
6834151.00 6609660.00 6480530.00 6393930.50 6329427.00 6278239.00 6235909.00 6199873.50 6168524.50 6140807.00
4.5. Analysis of the results When the shortage cost is only 100 TD, which is about the same cost as the cost of imported water, more than 40% of the total release in the average is supplied from groundwater. However, when p is multiplied by 7 or more, no groundwater is pumped in the average to supplement for the shortage of water. It is interesting to notice that in the worst case of supply (p 1000 TD in which case the expected total supply is only 56.44% of the expected supply when p is only 100 TD) the expected pro®t is only reduced by 10.1%. Also, if we compare the case where p 400 TD with the case where p 1000 TD, we can see that the expected release is reduced to 82%, while the reduction in expected pro®t is only of 4%. This shows that some substantial saving of groundwater may occur without aecting much the total expected pro®t. This would be the case if a careful allocation of the irrigation level for each selected crop is made. In fact, we can observe that the demand for wheat is satis®ed only at 56.5% in the worst case. Also, wheat is under irrigated in the average as soon as p exceeds 100 TD. However, the demand for sorghum is satis®ed at 78% in the worst case, and sorghum is under irrigated when p exceeds 200 TD. Finally, saower receives, in the worst case, 93% of its expected demand, and saf¯ower is under irrigated only when p exceeds 600 TD. This is mainly due to the fact that saower is considered more sensitive to water de®cit than the other two crops (saower would only provide 10% of maximum yield if not irrigated arti®cially at all,
wheat would provide 30% and sorghum 40%). In addition, saower generates by far more pro®t than the other two crops per thousand of cubic meter of irrigation water. Moreover, sorghum is more pro®table than wheat. Figs. 4±6 can be helpful in better understanding of these results. In these ®gures, we consider p 100 TD as the base case, and we denote groundwater by GW. From this illustrative example, we may conclude that it is possible to regulate pumping groundwater to avoid severe threats to the aquifer (such as total depletion, seawater intrusion and quality degradation) without much aecting the total pro®t. However, this may only be achieved if a careful selection of irrigation levels to dierent crops is made. This selection takes into account mainly the productivity of each crop as well as its sensitivity to the shortage of water. While this illustration is based on a hypothetical example, several parameters are real. This includes the yield response factors, some of the costs and the maximum yields of the dierent crops. Also, other parameters re¯ect to some extent the real situation (such as the sensitivity of the crops to the shortage of water and the linearity of the yield function). 5. Conclusion In this study, we deal with a special problem of the conjunctive use of ground and surface water. In this problem, both the supply and the demand are stochastic. Also, the aquifer is suering from
M.N. Azaiez, M. Hariga / European Journal of Operational Research 133 (2001) 653±666
663
Fig. 4. % of water supply from the ground stock as penalty cost varies.
Fig. 5. Water supply versus pro®t with the variation of the penalty cost.
Fig. 6. Irrigation levels of the crops as a percent of the mean demand.
severe overdrafts that may lead to total depletion of the aquifer, or to seawater intrusion and quality degradation. Therefore, we assume that pumping
groundwater would only be made in emergency situations. This can be achieved by placing important penalty fees for pumping groundwater. On
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the other hand, we allow de®cit irrigation. The operating policy consist on deciding on · how much of water to release from the main reservoir that receives a stochastic in¯ow, · how much groundwater to pump to supplement for the shortage of surface water supply, · how to allocate the total released water (possibly after supplementing with groundwater) among local reservoirs, · how to allocate water at local reservoirs among selected crops. We consider the problem for a single period, with a single decision-maker approach. We formulate the problem as a stochastic non-linear program. We show that this problem is concave with linear constraints. As a result, the KKT conditions are necessary and sucient to derive an optimal operating policy. We develop an algorithm to solve KKT conditions. We show some similarities between our problem and the stochastic two-echelon, single-period inventory problem. However, our problem has an added feature consisting of a stochastic supply to the main reservoir. We illustrate the model through a hypothetical example. In this illustration, the optimal operating policy shows that substantial groundwater saving may occur without aecting much the expected total pro®t. The example also shows how dierent crops may receive signi®cantly dierent proportions of their respective demand, in the case where de®cit irrigation is to be applied. This is due mainly to the sensitivity of crops to shortage of irrigation water, as well as to the productivity of each crop. Recently, Azaiez et al. (2001) have extended this problem to a planning horizon of several periods, which re¯ects much better the real situation. Another important extension would be to consider the problem in a multi-decision-maker framework, as the objectives of individual farmers and of water authorities may disagree. Also, cropping patterns in this problem are considered in a simplistic version, as we only decide on the irrigation level for pre-selected crops and pre-selected lands for each crop. The problem would be more realistic if cropping decisions are to be made to select among competing crops those that are the most rewarding. Also, land allocation to selected crops may be
made optimally taking into consideration the stochastic supply of water, the total land, the predecessors of competing crops, crop rotation, and other important factors to make the best exploitation of all input parameters. The reader is referred to Haouari and Azaiez (2001) for more details.
Acknowledgements The ®rst author was partially supported from a grant by ICARDA. He is also willing to extend his thanks to Professor M.S. Mattoussi from ``Faculte des Sciences Economique et Gestion, Tunis III'' for his assistance. Both authors also received partial support from the College of Engineering Research Center of King Saud University. Finally, the authors would like to thank two anonymous referees for their helpful suggestions, which improved the content and presentation of the paper. Appendix A. Illustration of Algorithm 3.1 When the water in¯ow is normally distributed, w N lw ; rw , and water demands have gamma distributions, gamma ai;j ; bi;j , the necessary and sucient conditions given in (10)±(12) become, respectively, R lw h k ; A:1 U hp hp Z Pi;j ai;j bi;j 1 G Di;j ; ai;j 1; bi;j dDi;j h ai;j 1 IRi;j Z 1 G Di;j ; ai;j ; bi;j dDi;j k h 0; IRi;j
ki n X X i1
Ai;j IRi;j
R 0;
A:2
A:3
j1
where U is the c.d.f. of the standard normal distribution, G ; a; b is the p.d.f. of the gamma distribution with shape and scale parameters a and
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b, respectively, and ai;j is the slope of the yield function Yi;j ai;j IRi;j =Di;j bi;j . Note that we assumed in the illustration that the initial stock of water is zero in all reservoirs (I Ii 0 for all 1 6 i 6 n). In the case of integer shape parameters (as for the illustration in Section 4), (A.2) can be rewritten as ai;j 2 k Pi;j ai;j bi;j X bi;j IRi;j e ai;j 1 k0 k! k
bi;j IRi;j e k!
bi;j IRi;j
bi;j IRi;j
h
aX i;j 1 k0
k
h 0;
or equivalently,
Pi;j ai;j bi;j h ai;j 1
aX i;j 2
a
bi;j IRi;j i;j ai;j 1!
k0 1
k
bi;j IRi;j e k!
k
bi;j IRi;j
hebi;j IRi;j 0:
h A:4
In Step 2 of Algorithm 3.1, we employed the following inverse normal function (Tijms, 1994): R
lw rw 2:51517 0:802853z 0:010328z2 z ; 1 1:432788z 0:189269z2 0:001308z3 A:5 where v " u 2 # u h k t : z Ln 1 hp
A:6
In addition, for a given k, we solved Eq. (A.4) using the bi-section method. Given the data reported in Tables 1±3, and a shortage cost of p 100 TD, we need to search for the optimal Lagrange-multiplier, k, over the interval 100; 25. Using steps 1±3 of Algorithm 3.1 repeatedly, the procedure converges after 22 iterations to k 99:425. Using (A.4)±(A.6), we get IRsg 1:113; IRwh 1:182; IRsf 2:048, and R 4523:46 (in 103 m3 ).
665
References Adig uzel, R., Cokunoglu, O., 1984. A descriptive decision process model for hierarchical management of interconnected reservoir systems. Water Resource Research 20 (7), 803±811. Aguado, E., Remson, I., 1980. Ground-water management with ®xed charges. Journal of the Water Resource Planning Management Division of the American Society of Civil Engineering 106, 375±382. Aguado, E., Remson, I., Pikul, M.F., Thomas, W.A., 1974. Optimum pumping for aquifer dewatering. Journal of Hydraulic Division of the American Society of Civil Engineering 100, 860±877. Azaiez, M.N., Hariga M., Al-Harkan, I., 2001. A chanceconstrained multi-period model for A multi-reservoir system under water de®cit and severe overdrafts of the Aquifer, submitted to JORS. Doorenbos, J., Kassam, A.H., 1979. Yield response to water. Irrigation and Drainage Paper 33, F.A.O., Rome. Dudley, J.N., 1988a. A single decision-maker approach to irrigation reservoir and farm management decision making. Water Resource Research 24 (5), 633±640. Dudley, J.N., 1988b. Volume sharing of reservoir water. Water Resource Research 24 (5), 641±648. Eiger, G., Shamir, U., 1991. Optimal operation of reservoirs by stochastic programming. Eng. Opt. 17, 293±312. English, M.J., 1990. De®cit irrigation. 1. Analytical framework. Journal of Irrigation and Drainage, Engineering Division, ASCE 116 (3), 399±412. English, M.J., Nuss, G.S., 1982. Designing for de®cit irrigation. Journal of Irrigation and Drainage, Engineering Division, ASCE 108 (2), 91±106. Gorelick, S.M., 1983. A review of distributed parameter groundwater management modeling methods. Water Resource Research 19, 305±319. Haouari, M., Azaiez, M.N., 2001. Optimal cropping patterns under water de®cit. European Journal of Operational Research 130 (1), 133±146. Hargraves, G.H., Samani, Z.A., 1984. Economic consideration of de®cit irrigation. Journal of Irrigation and Drainage Engineering Division, ASCE 110 (4), 343± 358. Loucks, D.P., Stedinger, J.R., Haith, D.A., 1981. Water Resource Systems Planning and Analysis. Prentice-Hall, Englewood Clis, NJ. Mannocchi, F., Mecarelli, P., 1994. Optimization analysis of de®cit irrigation systems. Journal of Irrigation and Drainage Engineering Division, ASCE 120 (3), 484± 503. Rao, N.H., Sharma, P.B.S., Chander, S., 1988. A simple dated water-production function for use in irrigated agriculture. Agricultural Water Management 13, 25±32. Remson, I., Gorelick, S.M., 1980. Management models incorporating ground-water variables. In: Yaron, D., Tapiero, C.S. (Eds.), Operations Research in Agriculture and Water Resources. North-Holland, Amsterdam.
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Stewart, J.I., Hagan, E.M., 1973. Functions to predict eects of crop water de®cits. Journal of Irrigation and Drainage Division, ASCE 99 (IR4), 421±439. Silver, E., Pyke, D., Peterson, R., 1998. Inventory Management and Production Planning and Scheduling, third ed. Wiley, New York. Tijms, H.C., 1994. Stochastic Models, An Algorithmic Approach. Wiley, New York. Vedula, S., Mujumdar, P.P., 1992. Optimal reservoir operation for irrigation of multiple crops. Water Resource Research 28 (1), 1±9. Vedula, S., Nagesh Kumar, D., 1996. An integrated model for optimal reservoir operation for irrigation of multiple crops. Water Resource Research 32 (4), 1101±1108. Wanakule, N., Mays, L., 1986. Optimal management of largescale aquifers: Methodology and applications. Water Resource Research 22 (4), 447±465.
Wang, D., Adams, B.J., 1986. Optimization of real-time reservoir operations with Markov decision processes. Water Resource Research 22 (3), 345±352. Willis, R., Liu, P., 1984. Optimization model for groundwater planning. Journal of Water Resource Planning Management Division of the American Society of Civil Engineering 110, 333±347. Willis, R., Newman, B.A., 1977. Management model for groundwater development. Journal of Water Resource Planning Management Division of the American Society of Civil Engineering 13, 159±171. Yeh, W.W.G., 1985. Reservoir management and operation models: A-state-of-the-art review. Water Resource Research 21 (12), 1797±1818.
www.elsevier.com/locate/dsw
Theory and Methodology
A single-period model for conjunctive use of ground and surface water under severe overdrafts and water de®cit M.N. Azaiez
*,1,
M. Hariga
Industrial Engineering Program, King Saud University, P.O. Box 800, Riyadh 11421, Saudi Arabia Received 19 July 1999; accepted 7 July 2000
Abstract We develop a model for a multi-reservoir system, where the in¯ow to the main reservoir and the demand for irrigation water at local areas are stochastic. The controlled releases from the main reservoir to local ones can be supplemented by groundwater from an aquifer suering from severe overdrafts. We consider high penalty costs for pumping groundwater in order to reduce the risk of total depletion of the aquifer as well as quality degradation and seawater intrusion. We analyze the problem for a single period with a single decision-maker approach. We allow de®cit irrigation in the problem of maximizing expected total pro®t for the entire region. We show that the formulated nonlinear stochastic problem is concave with linear constraints and propose an iterative procedure that generates an optimal operating policy. We illustrate the model through a hypothetical example. Ó 2001 Elsevier Science B.V. All rights reserved. Keywords: Reservoir management; De®cit irrigation; Stochastic supply; Stochastic demand
1. Introduction The region of interest, Mornag located at 20 km south of Tunis, initially imports water from the north of Tunisia to supplement for the shortage of groundwater, essentially the only source of irrigation water in the region. With the ever-increasing demand for irrigation water, overdrafts of * Corresponding author. Tel.: +966-1-4676828; fax: +966-14676652. E-mail address: [email protected] (M.N. Azaiez). 1 Currently on leave from Ecole Superieure de Commerce de Sfax, Tunisia.
groundwater have been observed during the last decades leading to serious threats of severe quality degradation and in some places to seawater intrusion, in addition to the risk of total depletion of the aquifer. To put some restrictions on groundwater withdrawals, the authorities established ``banning zones'' where it is disallowed to withdraw groundwater. At the same time, more water was imported from the north through the ``Medjerda-CapBon'' canal. Recently, several regions have been connected to this canal in order to reduce their respective local water de®cit. As a result, the canal became unable to totally ful®ll all the demands. Therefore, the supply of water from the
0377-2217/01/$ - see front matter Ó 2001 Elsevier Science B.V. All rights reserved. PII: S 0 3 7 7 - 2 2 1 7 ( 0 0 ) 0 0 2 1 2 - 5
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north has become stochastic according to the availability of water and to the aggregate demand of all regions connected to the canal. However, the local authorities in the region of interest plan to make the imported water as the main source of water supply and the groundwater as a supplemental source that can be used only when it is absolutely necessary. This plan is expected to occur at least for several seasons, until the ground stock is suciently replenished from natural and arti®cial recharges. This can be achieved by signi®cantly increasing the fees of withdrawing groundwater. The imported water to the region of interest is stored in a main reservoir. Then, releases are made to dierent local reservoirs to irrigate the local areas. The problem is to determine the optimal release policy from the main reservoir, possibly with supplemented groundwater, to local reservoirs and from local reservoirs to irrigated areas so that the total pro®t of the region is maximized. A stochastic supply of water from the north and a stochastic demand of water in each irrigated area characterize this special problem of conjunctive use of ground and (imported) surface water. The demand of irrigation water depends on prior rainfall and cropping decisions for the season or irrigation period. Such a period is usually of 8±10 weeks. Water al-
location from the main reservoir to the local ones is made at the beginning of each irrigation period. Therefore, water demand at local reservoirs is stochastic at the decision phase of water allocation from the main reservoir to the local ones. However, this demand is known at the irrigation phase. The uncertainty about the demand increases in the case of a multi-period allocation. In the present study, we are interested to analyze this problem from a single decision-maker approach (local authorities). We will restrict attention to a single season. The situation is depicted in Fig. 1. 2. Literature review There is a large number of published studies in reservoir management and operations models. These include Dudley (1988a,b), Wang and Adams (1986), Adig uzel and Cokunoglu (1984), Vedula and Mujumdar (1992), and Vedula and Nagesh Kumar (1996) to name only a few. Yeh (1985) provides a state-of-the-art review of reservoir management models. With the exception of few models, such as the one by Eiger and Shamir (1991), most of reservoir management models consider only the supply or the demand as stochastic. This is due to the added
Fig. 1. A description of the problem.
M.N. Azaiez, M. Hariga / European Journal of Operational Research 133 (2001) 653±666
complexities by treating both demand and supply as stochastic. Also, as for the present study, most of the models developed for reservoir operations for irrigation consider a single decision-maker approach (e.g., Dudley, 1988a; Vedula and Mujumdar, 1992; Vedula and Nagesh Kumar, 1996; Haouari and Azaiez, 2001). Dudley (1988b), however, develops a model for irrigation reservoir and farm management with a multi-decision-maker approach, namely a reservoir manager and individual irrigation farmers. A lot of eort has been devoted to groundwater management (for instance, Aguado et al., 1974; Aguado and Remson, 1980; Willis and Newman, 1977; Remson and Gorelick, 1980; Willis and Liu, 1984; Wanakule and Mays, 1986). Many of the groundwater management models use simulation combined with optimization, resulting in the socalled simulation±management models (Wanakule and Mays, 1986). Gorelick (1983) reviews the literature for such models. De®cit irrigation is gaining interest in irrigation planning. A number of models have been developed in the literature, where crops are deliberately under irrigated in order to save water and increase the irrigated area and possibly the pro®t. Stewart and Hagan (1973) develop a multiplicative formula for crop yield as a function of applied irrigation water. This formula is widely used in de®cit irrigation. Doorenbos and Kassam (1979) analyze experimental data on crop yield response to water, and empirically derive yield response factors Ki . These factors are sensitivity indices for water stress in the speci®ed growth stage i of the crop. Rao et al. (1988) compare three dated water-production function models and conclude that a simple heuristic multiplicative form is applicable over a wide range of stress conditions. Vedula and Mujumdar (1992) and Mannocchi and Mecarelli (1994) develop procedures to assess crop yields as a function of applied water, in terms of the ratio of actual to maximum evapotranspiration. 3. The model The main reservoir receives a stochastic quantity of water from the north. The decision-
655
maker pays a price proportional to the quantity received, and allocates an amount of water to each local reservoir. If the total amount of water to be allocated to these local reservoirs exceeds the active capacity of the main reservoir, then an emergency withdrawal from the aquifer is applied to supplement for the shortage. Otherwise, the amount of water remaining in the main reservoir will be maintained for future uses. The decisionmaker will also maintain surplus water at each local reservoir if the obtained total stock of water exceeds the local demand. This action, however, would be unlikely to happen due to the constant de®cit of irrigation water in the region. If a shortage occurs, then this would be re¯ected in the response to yield for the period of interest. We will neglect the delivery costs from the main reservoir to the local ones, as these costs are minor. The problem is to identify the optimal operating policy that permits the maximization of the total contribution to the pro®t during the period of interest. An operating policy consists of three decision stages: 1. Identi®cation of the appropriate amount of water to be allocated to the entire region. 2. Allocation of this water optimally among the dierent local reservoirs for the period of interest. 3. Allocation of water at local reservoirs optimally among the selected crops. In this paper, we are interested to analyze the problem for a single period (or irrigation season). Recently, Azaiez et al. (2001) extend the problem to a planning horizon of several periods. Other possible extensions would include a multi-decision-maker approach, involving the authorities and the farmers. Our single-period problem bears some similarities with the stochastic two-echelon, singleperiod inventory problem (see Silver et al., 1998), where the main reservoir represents the central warehouse and local reservoirs represent the retailers. However, our problem has an added feature consisting of a stochastic supply to the main reservoir (the central warehouse in the inventory problem). In this analogy between the stochastic two-echelon, single-period inventory problem and
656
M.N. Azaiez, M. Hariga / European Journal of Operational Research 133 (2001) 653±666
the current problem of conjunctive use of ground and surface water, the holding costs have two components, direct and opportunity costs. Direct costs are represented by the costs of maintaining water in the dierent reservoirs for future uses (chlorinating and inspection). Opportunity costs consist of over-allocating water at the expense of reducing other users' activities. These opportunity costs are very important if the over-allocation has totally or partially been supplied from an emergency withdrawal of groundwater. To reduce the computational burden, we will assume that such holding costs are linear. Also, shortage cost consists of the important fee of an emergency withdrawal from the aquifer. Another indirect shortage cost, incurred at local reservoirs, is re¯ected in the reduced yield of the crops to be grown as a result of de®cit irrigation. This can be considered as an opportunity cost. No salvage cost is assumed. 3.1. Notation and assumptions The in¯ow of water to the main reservoir for the period of interest is stochastic. Therefore, we set the following: w: the stochastic in¯ow to the main reservoir, fw : the probability distribution function (p.d.f.) of w, Fw : the cumulative distribution function (c.d.f.) of w, lw : the expected value of the random variable w. The demand for water of each selected crop at a local reservoir is also stochastic. Therefore, we also set for each local reservoir i 1 6 i 6 n, where n is the number of local reservoirs), and for each crop j (1 6 j 6 ki , where ki is the number of selected crops to be grown on local area i): Di;j : the stochastic demand for water of crop j per unit of area at local area i for the period of interest, fi;j : the p.d.f. of Di;j , Fi;j : the c.d.f. of Di;j . Since the decision-maker will decide on the total water to be allocated (release from main reservoir combined possibly with groundwater), on the share of each reservoir i, as well as on the ir-
rigation level to be assigned to each selected crop, we let the decision variables be: R: total water to be released from the main reservoir, Ri : water to be released to reservoir i (share of reservoir i) for 1 6 i 6 n, IRi;j : irrigation level per unit of area to be allocated to crop j at local area i for 1 6 j 6 ki ; 1 6 i 6 n. The cost parameters are given by: C0 : unit cost of water imported from the north, p: unit cost of an emergency order (penalty for withdrawing groundwater, expressing the shortage cost), h: unit holding cost at the main reservoir (cost of maintaining water for future use), hi : unit holding cost of maintaining reservoir i for 1 6 i 6 n. Moreover, we let I: initial amount of water in the main reservoir before receiving the stochastic quantity of water from the north, Ii : initial amount of water in the ith reservoir before receiving its share from the main reservoir, Yi;j IRi;j ; Di;j : the expected yield of crop j per unit area at reservoir i, given some demand per unit area Di;j , during the period of interest, Ai;j : area allocated to crop j in local area i, Pi;j : pro®t per unit area of crop j grown at local area i. We assume that Yi;j IRi;j ; Di;j is concave and twice dierentiable. This assumption is not restrictive given the typical shape of Yi;j IRi;j ; Di;j as a function of applied irrigation water. This shape is given in Fig. 3. We also assume that area Ai;j is allocated to crop j independently of the amount of water to be made available. This can be justi®ed, for instance, from the nature of the soil, rotation constraints and input factors. Thus, the decision variables must satisfy n X
Ri R:
1
i1 ki X j1
Ai;j IRi;j Ii Ri
81 6 i 6 n:
2
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657
3.2. The cost function
3.3. The contribution to the yield
We will consider the following components of the expected total cost for the period of interest: 1. expected cost of imported water is equal to
The contribution to the yield during the period of interest of a given selected crop j at the ith reservoir will depend on the irrigation level to be allocated to this crop. This irrigation level would depend on the demand for water of the crop, and would be constrained by the initial stock, Ii , and the share, Ri , as given in (2). It is reasonable to consider the contribution to the yield as a predetermined fraction of the total yield. This can be justi®ed from Stewart's multiplicative formula (see for example, Mannocchi and Mecarelli, 1994). Alternatively, we may consider that the period of interest is the only dry period, in which shortage of irrigation water may occur. Consequently, the period of interest would be responsible for any reduction in yield that may occur. We would adopt this latter assumption for the present study. Generally speaking, the relationship between crop yield and applied irrigation water follows a trend similar to the one provided in Fig. 2. Similar relationships between crop yield and applied water can be found in English and Nuss (1982), English (1990), and Hargraves and Samani (1984). The ®gure points out two zones corresponding, respectively, to under and over irrigation of the crop. In the ®rst zone, corresponding to de®cit type of irrigation, the relationship between applied water and crop yield is approximately linear. The second zone, characterized by the beginning of the greatest curvature in the function, indicates a diminution in the rate of increase of the crop yield due to the negative eects of soil water surplus. The contribution to the yield or the yield function for the present study will be similar to the one above in case of de®cit irrigation (®rst zone). However, if the amount of water assigned to a
C0 lw ; 2. expected holding costs at the main reservoir is equal to Z h
1 R I
w I
Rfw w dw;
3. expected shortage cost at the main reservoir is equal to Z p
R I 0
R
I
wfw w dw;
4. expected holding costs at the ith reservoir is equal to hi
ki X
Z Ai;j
IRi;j 0
j1
IRi;j
Di;j fi;j Di;j dDi;j :
In fact, as we will explain below, the operating policy will not allow a crop to be over irrigated. Therefore, the surplus of allocated water to each crop (if any) will remain in the local reservoir for future use. From the individual costs above, the expected total cost will be ETC R; IRi;j ; 1 6 i 6 n; 1 6 j 6 ki Z 1 C0 lw h w I Rfw w dw Z p Z
0
R I
R I
0 IRi;j
R
I
wfw w dw
n X i1
IRi;j
Di;j fi;j Di;j dDi;j :
hi
ki X
Ai;j
j1
3
It can be easily shown that the total expected cost ETC R; IRi;j ; 1 6 i 6 n; 1 6 j 6 ki is a convex function. Note that the shortage cost of not meeting the demand at the ith reservoir will be re¯ected later in the contribution to the yield.
Fig. 2. Crop yield as a function of applied water.
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M.N. Azaiez, M. Hariga / European Journal of Operational Research 133 (2001) 653±666
given crop exceeds the demand, then the surplus of water will remain in the reservoir. Therefore, the yield function per unit of area in the second zone will be of the form Yi;j IRi;j ; Di;j
Yi;jmax
if Di;j 6 IRi;j :
4
In other words, the function Yi;j IRi;j ; Di;j will be increasing if IRi;j < Di;j , as in the ®rst zone of Fig. 2 and will be constant in IRi;j if IRi;j P Di;j . Thus, the shape of the function Yi;j IRi;j ; Di;j will be similar to the one given in Fig. 3. For a given crop j at local area i, the expected return generated by some assigned irrigation level IRi;j is given by Z 1 Pi;j Ai;j Yi;j IRi;j ; Di;j fi;j Di;j dDi;j ER IRi;j 0 ( Pi;j Ai;j Fi;j IRi;j Yi;jmax Z
1 IRi;j
) Yi;j IRi;j ; Di;j fi;j Di;j dDi;j : 5
Proposition 3.1. ER IRi;j is concave. Proof.
( dER IRi;j Pi;j Ai;j fi;j IRi;j Yi;jmax dIRi;j Z 1 dYi;j IRi;j ; Di;j fi;j Di;j dDi;j dIR i;j IRi;j ) Yi;j IRi;j ; IRi;j fi;j IRi;j :
6
However, Yi;j IRi;j ; IRi;j Yi;jmax , therefore, dER IRi;j dIRi;j
Z
Pi;j Ai;j
1
dYi;j IRi;j ; Di;j fi;j Di;j dDi;j : dIRi;j
IRi;j
7 It follows that d2 ER IRi;j dIR2i;j (Z
d2 Yi;j IRi;j ; Di;j fi;j Di;j dDi;j 2 IRi;j dIRi;j ) dYi;j IRi;j ; IRi;j fi;j IRi;j : dIRi;j
Pi;j Ai;j
1
Note that the second term in the right-hand side vanishes as the function Yi;j ; IRi;j is constantly equal to Yi;jmax . It is clear therefore that the ®nal expression of the second derivative of ER is nonpositive from the concavity of Yi;j and the nonnegativity of fi;j . The global optimization program of total water release from main reservoir and allocation among local reservoirs and then among selected crops is given by max
R;Ri ;IRi;j
ki n X X i1
ER IRi;j
ETC R; IRi;j ;
j1
1 6 i 6 n; 1 6 j 6 ki s:t:
ki X
Ai;j IRi;j
Ri I i ;
1 6 i 6 n;
j1 n X
Ri
R 0;
i1
R; Ri ; IRi;j P 0;
1 6 i 6 n; 1 6 j 6 ki : 8
Fig. 3. The function Yi;j IRi;j ; Di;j .
It follows from the concavity of the expected return functions, ER IRi;j , and the convexity of the expected total cost function, ETC R; IRi;j ; 1 6 i 6 n; 1 6 j 6 ki that this global program is a concave program with linear constraints. Therefore, Karush±Kuhn±Tucker (KKT) conditions are
M.N. Azaiez, M. Hariga / European Journal of Operational Research 133 (2001) 653±666
necessary and sucient for optimality. To reduce the computational eort in solving for the KKT conditions, we replace the dependent constraints ki X
Ai;j IRi;j
Ri I i ;
Ri
i1
R0
i1
j1
by the single constraint n X
ki X
i1
j1
Ai;j IRi;j
R
n X
g k
Ii :
Therefore, the KKT conditions will be given by h
"
hi Fi;j IRi;j
k60
for 1 6 i 6 n; 1 6 j 6 ki ; # ki n X n X X Ai;j IRi;j Ii ; R j1
i1
R h
"
IRi;j
i1
p hFw R I k 0; Z 1 oYi;j IRi;j ; Di;j Pi;j oIRi;j IRi;j
fi;j Di;j dDi;j
hi Fi;j IRi;j k 0
for 1 6 i 6 n; 1 6 j 6 ki ; R P 0;
IRi;j P 0 for 1 6 i 6 n; 1 6 j 6 ki ; 9
where k is the Lagrange-multiplier associated with the single dependent constraint. After ignoring the non-negativity constraints of the optimization problem given by (8), the KKT conditions reduce to Fw R Z Pi;j
I 1 IRi;j
h k ; hp
10
oYi;j IRi;j ; Di;j fi;j Di;j dDi;j oIRi;j
hi Fi;j IRi;j k 0;
R
n X
Ii :
12
i1
ki n X X i1
i1
p hFw R I k 6 0; Z 1 oYi;j IRi;j ; Di;j Pi;j fi;j Di;j dDi;j oIRi;j IRi;j
Ai;j IRi;j
From (10), it is easy to show that a feasible k must belong to the interval p; h. Moreover, assuming that (10) and (11) can be solved for R and IRi;j , respectively, as a function of k, then it is possible to show that the following one-dimensional function:
1 6 i 6 n;
j1 n X
ki n X X
659
1 6 i 6 n; 1 6 j 6 ki ; 11
j1
Ai;j IRi;j k
R k
n X
Ii
i1
is strictly increasing. Based on these results, we provide the following line-search algorithm to solve (10)±(12). Algorithm 3.1. Step 0. Select some small real number e > 0, and set L p and U h. Step 1. Set k L U =2. Step 2. Use (10) and (11) to ®nd R k; and IRi;j k. Step 3. If jg kj < e, then stop. Else if g k > 0, then set U k and go to step 1. Else g k < 0, then set L k and go to step 1. End. We will repeatedly use Algorithm 3.1 to derive optimal operating policies in the example below. In addition, we provide a detailed illustration on the use of Algorithm 3.1 in Appendix A. In the illustration below, Algorithm 3.1 is applied to a small problem. However, our computational experience with this algorithm indicates that it is ecient for large-scale problems.
4. Illustration We will provide a simple example to show the applicability of the model. This example is mostly based on hypothetical data. This is due to the fact that real data collection is still at its very ®rst stage by now. However, we do not expect that the data
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M.N. Azaiez, M. Hariga / European Journal of Operational Research 133 (2001) 653±666
Table 1 Distribution of selected crops in the three local areas Crop Sorghum Wheat Saower Total
Area (ha) Area 1
Area 2
Area 3
600 400 ± 1000
300 ± 700 1000
± 500 500 1000
to be used would be signi®cantly dierent from the true situation. We consider three local areas n 3, each having a total land of 1000 hectares (ha). We assume that the selected crops to be grown in these areas are given by Table 1.
4.1. The demand and supply distributions The families of normal and gamma distributions have found frequent use in water resource planning (Loucks et al., 1981). The normal distribution, as a symmetric distribution, does not always satisfactorily model phenomena such as stream¯ows, which are necessarily positive and have skewed distributions. The lognormal distribution usually ®ts better for stream¯ows. However, within a same period (for instance a wet period) the normal distribution may reasonably ®t for stream¯ows, demand for irrigation water and rainfalls. The gamma distribution, which arises naturally in many problems in hydrology, has long been used to model daily, monthly and annual stream¯ows. It also has a very reasonable shape to represent rainfall. The amount of water to be imported from the north to the region of interest depends on rainfalls in the north as well as on the demand for water in
the other regions that are connected to the canal. We will assume that the stochastic in¯ow, w, to the main reservoir for the speci®c period of interest can adequately be represented by a normal distribution. We will also use the gamma distribution to represent the demand for irrigation water in the three local areas considered in this illustration. The details are provided in Table 2. In Table 2, the parameters given for the normal distribution are the mean and the variance, and for the gamma distribution the shape and scale parameters, respectively. Note that we have considered the demand of a particular crop as independent of the local area in which the crop will be grown. 4.2. The costs We will arbitrarily assign the holding costs (for maintaining water) to be 25 Tunisian dinars (TD) in the main reservoir and 30 TD in each of the local reservoirs per thousand of cubic meters for the period of interest. The purchasing cost is 90 TD per thousand of cubic meters. We will analyze the problem for a variety of shortage costs ranging from 100 to 1000 TD per thousand of cubic meters. 4.3. The returns We set the returns in TD per tonne to be, respectively, 150, 200, and 300 for sorghum, wheat, and saower. 4.4. The yield functions If a particular crop has s growth stages, for some integer s (usually s is taken to be between 3
Table 2 Demand and supply distributions Random variable
Distribution
Mean
w D1 (Sorghum per ha) D2 (Wheat per ha) D3 (Saower per ha)
Normal 2:7; 0:72 Gamma (2.0; 2.5) Gamma (2.0; 2.0) Gamma (2.0; 1.5)
2.70 0.80 1.00 1.33
(millions of m3 ) (103 m3 /ha) (103 m3 /ha) (103 m3 /ha)
Variance 0.49 0.32 0.50 0.89
M.N. Azaiez, M. Hariga / European Journal of Operational Research 133 (2001) 653±666
and 5), then the fraction of the actual yield Ya to the maximum yield Ym , due to de®cit irrigation, can be approximated by (Mannocchi and Mecarelli, 1994) Ya Ym
s Y
1
i1
Kyi 1
ETa ETm
;
13
where i is a generic growth stage, Kyi is the yield response factor of the crop at growth stage i, ETa and ETm are, respectively, the actual and maximum evapotranspiration. In the present study, since we are only analyzing the one-growth stage case, we assume that de®cit irrigation would only occur for the particular period of interest (say the dry period). In this case, no reduction in yield would occur as a result of de®cit irrigation for the remaining growth stages and the yield function would be of the form Ya ETa 1 Ky 1 : 14 Ym ETm The fraction of the actual to the maximum evapotranspiration depends on the irrigation level to be applied compared with the irrigation demand of the crop. This dependence is approximately linear, as was illustrated in Figs. 2 and 3. If no irrigation water is applied, then ETa will result from rainfall only and the fraction of ETa =ETm will have some value d (with 0 < d < 1) that can usually be estimated empirically. For the purpose of this illustration, we will assign values for d arbitrarily (but re¯ecting the sensitivity of each crop to the shortage of arti®cial irrigation). If, however, enough water is applied to satisfy full water requirement of the crop, then ETa will simply
661
coincide with ETm . Therefore, the fraction ETa =ETm will be approximated by ETa 1 dIR d: D ETm
15
Therefore, the yield function will be approximated by Ya Ym 1
Ky 1
d 1
IR D
:
16
We assume that the period of interest is the ¯owering stage. Consequently, we are using the corresponding yield response factors. The yield function and its dierent parameters are summarized in Table 3. In Table 3, wh stands for wheat, sg for sorghum and sf for saower. Note that there is no need to specify the local area in which each crop will be grown as we assumed that this would not aect the expected yield and returns. The optimal operating policy as well as the corresponding expected pro®t are given in Table 4 for a range of shortage costs. In Table 4, the shortage cost, p, as well as all other costs, pro®ts and returns are in TD. Also, R (the total release) is in thousands of cubic meters, and IRsg ; IRwh , and IRsf (the irrigation levels per unit of area allocated to sorghum, wheat, and saower, respectively) are in thousands of cubic meters per hectare. In addition, ETR is the expected total return for the entire region, MR cost is the expected cost at main reservoir (expected holding plus shortage costs), LR costs are the aggregate expected holding costs at the local reservoirs, and Pro®t is the expected total pro®t of the region ETR ETC.
Table 3 Yields and yield response factors of the selected crops Parameters
Crop Sorghum
Wheat
Saower
Ky d Ym (t/ha) Ya (t/ha)
0.55 0.4 16 5.28 IRsg =Dsg +10.72
0.65 0.3 7 3.185 IRwh =Dwh +3.815
0.55 0.1 12 5.94 IRsf =Dsf +6.06
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M.N. Azaiez, M. Hariga / European Journal of Operational Research 133 (2001) 653±666
Table 4 Optimal operating policy for a range of shortage costs p
R
IRsg
IRwh
IRsf
ETR
MR cost
LR cost
Pro®t
100 200 300 400 500 600 700 800 900 1000
4523.460 3708.291 3324.697 3099.604 2946.232 2832.069 2742.165 2668.568 2606.533 2553.107
1.113 0.911 0.816 0.760 0.722 0.693 0.671 0.653 0.637 0.624
1.182 0.926 0.806 0.736 0.688 0.652 0.624 0.601 0.582 0.565
2.048 1.712 1.554 1.461 1.398 1.351 1.314 1.283 1.258 1.236
7304183.50 7089560.00 6957981.00 6869941.00 6804943.00 6753782.00 6711764.00 6676199.50 6645383.00 6618222.50
182465.00 206943.50 210581.47 212438.08 214065.27 215602.52 217060.77 218441.36 219720.92 220906.20
44567.00 29956.50 23869.50 20572.50 18450.50 16940.50 15794.00 14884.50 14137.50 13509.50
6834151.00 6609660.00 6480530.00 6393930.50 6329427.00 6278239.00 6235909.00 6199873.50 6168524.50 6140807.00
4.5. Analysis of the results When the shortage cost is only 100 TD, which is about the same cost as the cost of imported water, more than 40% of the total release in the average is supplied from groundwater. However, when p is multiplied by 7 or more, no groundwater is pumped in the average to supplement for the shortage of water. It is interesting to notice that in the worst case of supply (p 1000 TD in which case the expected total supply is only 56.44% of the expected supply when p is only 100 TD) the expected pro®t is only reduced by 10.1%. Also, if we compare the case where p 400 TD with the case where p 1000 TD, we can see that the expected release is reduced to 82%, while the reduction in expected pro®t is only of 4%. This shows that some substantial saving of groundwater may occur without aecting much the total expected pro®t. This would be the case if a careful allocation of the irrigation level for each selected crop is made. In fact, we can observe that the demand for wheat is satis®ed only at 56.5% in the worst case. Also, wheat is under irrigated in the average as soon as p exceeds 100 TD. However, the demand for sorghum is satis®ed at 78% in the worst case, and sorghum is under irrigated when p exceeds 200 TD. Finally, saower receives, in the worst case, 93% of its expected demand, and saf¯ower is under irrigated only when p exceeds 600 TD. This is mainly due to the fact that saower is considered more sensitive to water de®cit than the other two crops (saower would only provide 10% of maximum yield if not irrigated arti®cially at all,
wheat would provide 30% and sorghum 40%). In addition, saower generates by far more pro®t than the other two crops per thousand of cubic meter of irrigation water. Moreover, sorghum is more pro®table than wheat. Figs. 4±6 can be helpful in better understanding of these results. In these ®gures, we consider p 100 TD as the base case, and we denote groundwater by GW. From this illustrative example, we may conclude that it is possible to regulate pumping groundwater to avoid severe threats to the aquifer (such as total depletion, seawater intrusion and quality degradation) without much aecting the total pro®t. However, this may only be achieved if a careful selection of irrigation levels to dierent crops is made. This selection takes into account mainly the productivity of each crop as well as its sensitivity to the shortage of water. While this illustration is based on a hypothetical example, several parameters are real. This includes the yield response factors, some of the costs and the maximum yields of the dierent crops. Also, other parameters re¯ect to some extent the real situation (such as the sensitivity of the crops to the shortage of water and the linearity of the yield function). 5. Conclusion In this study, we deal with a special problem of the conjunctive use of ground and surface water. In this problem, both the supply and the demand are stochastic. Also, the aquifer is suering from
M.N. Azaiez, M. Hariga / European Journal of Operational Research 133 (2001) 653±666
663
Fig. 4. % of water supply from the ground stock as penalty cost varies.
Fig. 5. Water supply versus pro®t with the variation of the penalty cost.
Fig. 6. Irrigation levels of the crops as a percent of the mean demand.
severe overdrafts that may lead to total depletion of the aquifer, or to seawater intrusion and quality degradation. Therefore, we assume that pumping
groundwater would only be made in emergency situations. This can be achieved by placing important penalty fees for pumping groundwater. On
664
M.N. Azaiez, M. Hariga / European Journal of Operational Research 133 (2001) 653±666
the other hand, we allow de®cit irrigation. The operating policy consist on deciding on · how much of water to release from the main reservoir that receives a stochastic in¯ow, · how much groundwater to pump to supplement for the shortage of surface water supply, · how to allocate the total released water (possibly after supplementing with groundwater) among local reservoirs, · how to allocate water at local reservoirs among selected crops. We consider the problem for a single period, with a single decision-maker approach. We formulate the problem as a stochastic non-linear program. We show that this problem is concave with linear constraints. As a result, the KKT conditions are necessary and sucient to derive an optimal operating policy. We develop an algorithm to solve KKT conditions. We show some similarities between our problem and the stochastic two-echelon, single-period inventory problem. However, our problem has an added feature consisting of a stochastic supply to the main reservoir. We illustrate the model through a hypothetical example. In this illustration, the optimal operating policy shows that substantial groundwater saving may occur without aecting much the expected total pro®t. The example also shows how dierent crops may receive signi®cantly dierent proportions of their respective demand, in the case where de®cit irrigation is to be applied. This is due mainly to the sensitivity of crops to shortage of irrigation water, as well as to the productivity of each crop. Recently, Azaiez et al. (2001) have extended this problem to a planning horizon of several periods, which re¯ects much better the real situation. Another important extension would be to consider the problem in a multi-decision-maker framework, as the objectives of individual farmers and of water authorities may disagree. Also, cropping patterns in this problem are considered in a simplistic version, as we only decide on the irrigation level for pre-selected crops and pre-selected lands for each crop. The problem would be more realistic if cropping decisions are to be made to select among competing crops those that are the most rewarding. Also, land allocation to selected crops may be
made optimally taking into consideration the stochastic supply of water, the total land, the predecessors of competing crops, crop rotation, and other important factors to make the best exploitation of all input parameters. The reader is referred to Haouari and Azaiez (2001) for more details.
Acknowledgements The ®rst author was partially supported from a grant by ICARDA. He is also willing to extend his thanks to Professor M.S. Mattoussi from ``Faculte des Sciences Economique et Gestion, Tunis III'' for his assistance. Both authors also received partial support from the College of Engineering Research Center of King Saud University. Finally, the authors would like to thank two anonymous referees for their helpful suggestions, which improved the content and presentation of the paper. Appendix A. Illustration of Algorithm 3.1 When the water in¯ow is normally distributed, w N lw ; rw , and water demands have gamma distributions, gamma ai;j ; bi;j , the necessary and sucient conditions given in (10)±(12) become, respectively, R lw h k ; A:1 U hp hp Z Pi;j ai;j bi;j 1 G Di;j ; ai;j 1; bi;j dDi;j h ai;j 1 IRi;j Z 1 G Di;j ; ai;j ; bi;j dDi;j k h 0; IRi;j
ki n X X i1
Ai;j IRi;j
R 0;
A:2
A:3
j1
where U is the c.d.f. of the standard normal distribution, G ; a; b is the p.d.f. of the gamma distribution with shape and scale parameters a and
M.N. Azaiez, M. Hariga / European Journal of Operational Research 133 (2001) 653±666
b, respectively, and ai;j is the slope of the yield function Yi;j ai;j IRi;j =Di;j bi;j . Note that we assumed in the illustration that the initial stock of water is zero in all reservoirs (I Ii 0 for all 1 6 i 6 n). In the case of integer shape parameters (as for the illustration in Section 4), (A.2) can be rewritten as ai;j 2 k Pi;j ai;j bi;j X bi;j IRi;j e ai;j 1 k0 k! k
bi;j IRi;j e k!
bi;j IRi;j
bi;j IRi;j
h
aX i;j 1 k0
k
h 0;
or equivalently,
Pi;j ai;j bi;j h ai;j 1
aX i;j 2
a
bi;j IRi;j i;j ai;j 1!
k0 1
k
bi;j IRi;j e k!
k
bi;j IRi;j
hebi;j IRi;j 0:
h A:4
In Step 2 of Algorithm 3.1, we employed the following inverse normal function (Tijms, 1994): R
lw rw 2:51517 0:802853z 0:010328z2 z ; 1 1:432788z 0:189269z2 0:001308z3 A:5 where v " u 2 # u h k t : z Ln 1 hp
A:6
In addition, for a given k, we solved Eq. (A.4) using the bi-section method. Given the data reported in Tables 1±3, and a shortage cost of p 100 TD, we need to search for the optimal Lagrange-multiplier, k, over the interval 100; 25. Using steps 1±3 of Algorithm 3.1 repeatedly, the procedure converges after 22 iterations to k 99:425. Using (A.4)±(A.6), we get IRsg 1:113; IRwh 1:182; IRsf 2:048, and R 4523:46 (in 103 m3 ).
665
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