An integrated model for the development of marginal water sources in the Negev Desert

ELSEVIER

EUROPEAN JOURNAL OF OPERATIONAL RESEARCH European Journal of Operational Research 81 (•995) 35-49

Case Study

An integrated model for the development of marginal water sources in the Negev Desert " Jack Brimberg a,,, Abraham Mehrez b,c, Gideon Oron

c

a Department of Engineering Management, Royal Military College of Canada, Kingston, Ont., Canada, K7K 5LO b Graduate School of Management, Kent State University, Kent, OH 44242, USA c The Blaustein Institute for Desert Research and Department of Industrial Engineering and Management, Ben-Gurion University of the Negev, Kiryat Sdeboker, Israel

Received February 1994

Abstract This work reports on a mathematical model dealing with the optimal development of marginal water sources in the Negev Desert, a region located in the south of Israel, inhabited by less than one-tenth of the whole population of Israel, and consisting of about half of its area. A decomposable mixed linear zero one integer programming problem is formulated and analyzed to integrate decision-making at the local and regional levels. At the regional level, the problem is one of allocating a limited supply of high quality water from a primary regional source among a number of local demand sites. Each demand site must then determine the optimal investment strategy for developing its local marginal water sources in order to make up the deficit in supply and meet net water quality and budgetary requirements at the site. The relationship between the local and regional levels of decision-making is studied, and a procedure to solve the combined problem is proposed. A small case study is presented to illustrate the model.

1. Introduction The main supply of water to the Negev region comes from the National Water Carrier (NWC). This system pumps water from the Sea of Galilee and the coastal and mountain aquifers in the northern and central areas of the country, to the drier regions in the South. The motivation for integrating water supply through the N W C that was constructed in the fifties was to satisfy the needs of the whole country for the foreseeable future. However, a period of drought coupled with increasing demand in all sectors of the economy has resulted in overpumpage in the water supply system in recent years. Thus, it is agreed (see Nativ and Issar [5]) that the current system is not capable to expand much further to supply expected water needs toward the end of the century, not speak beyond the year 2000. In order to reduce the demands on the N W C and alleviate the problems associated with over-pumpage, it has become necessary to develop the marginal water sources existing in the Negev. Water consumption in the Negev is

* This case study was accepted by Helman Stern. * Corresponding author. 0377-2217/95/$09.50 © 1995 Elsevier Science B.V. All rights reserved SSDI 0377-2217(94)00052-E

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J. Brimberg et al. /European Journal of Operational Research 81 (1995) 35-49

principally for agricultural purposes, and the cost of supply from the NWC is subsidized by the government. Thus, the development of local marginal sources would also allow the diversion of the high-quality water from the NWC to more sensitive uses in the industrial and domestic sectors. The following types of marginal water sources are considered by the model: (i) Treated Wastewater (TWW). Characterized by high investment costs in plant and equipment and the dual piping system required to distribute it separately from potable water. For small settlements this source will be of very limited capacity. (ii) Runoff water (ROW). This source derives from the sporadic rainfalls which occur in the Negev only during the winter season. Characterized by high capital investment costs associated with collection and storage, and by high quality. (iii) Saline groundwater (SGW). This source occurs in deep aquifers (as much as 1000 m) throughout the country and particularly in the Negev. The disadvantages associated with SGW are the high variable costs of pumpage and its inherent low quality. However, mixing with higher quality sources will extend the potential use of this source (e.g., see Letey and Dinar [4] and Pasternak et al. [6]). In the next section, a mixed linear, binary integer programming problem is formulated to consolidate decision-making at the local and regional levels. This involves the optimal allocation to demand sites of a limited supply from the NWC (the regional source) while simultaneously developing the most economical marginal sources at the local sites. Also, increasing pressure on the price charged by the NWC may result in alternative sources being economically feasible irrespective of supply constraints. The model minimizes total annual variable costs and capital recovery costs, subject to a regional supply constraint on the NWC, capacity constraints on the annual amount of water obtainable from each marginal source developed at a site only for its own consumption, and quality and demand constraints at each site. The demand at a site is given by its total requirement less the supply from existing local sources (e.g., desalination plants in Eilat). The global problem is decomposed into a set of small local problems to study the relation between the two levels of decision-making. Taking advantage of the simple structure of the local problem, a sensitivity analysis is employed to derive a strategic view integrating the local and regional decision-making levels using the Lagrangian relaxation technique (see for example Fisher [3]). In Section 3, the local problem is formulated at the operational level as a linear programming problem. For illustrative purposes, the case where all marginal sources are developed at the local site is analyzed for a varying level of water supplied from the NWC. The analysis of the operational problem is further extended in this section to deal with the capital budgeting aspect at the local level. In Section 4 the problem of allocating water from the NWC is discussed, and a Lagrangian relaxation method is suggested to solve it. A small case study and conclusions are provided in the final sections. Although final decisions have not been undertaken by the government of Israel, regional decision makers and experts pursue the enclosed study, dealing with the economic optimization aspects of developing new water sources in the Negev. The collection of data to analyze the implications of the suggested model may take a while, so that only the results of a pilot analysis are reported herel However, we believe that our model and the analytical results presented will enlighten the decision-making process aimed to develop water resources in the Negev, as well as in other arid regions. (Note that for regional quality management problems, the reader is referred to Schwarz et al. [8], and on the general management of water resources systems, see Willis and Yeh [9] and Yeh [10].)

2. A static model for local and regional planning

The main objective of the paper is to formulate a model which adequately describes the essential features of the problem, and provides a basic tool for decision-makers in planning the development of

J. Brimberg et al. ~European Journal of Operational Research 81 (1995) 35-49

37

marginal sources under various limits of supply from the NWC. Thus, it was considered sufficient at this stage to aggregate the level of detail in the model on an annual basis. In addition, the problem was linearized by neglecting higher-order effects, such as the quality of water on crop yields (and hence agricultural productivity and related profits). This approach is reasonable when the acceptable range of water quality applied (net quality when blending is considered) at the sites is sufficiently tight. For practical implementation at a later stage, a more detailed approach should be considered, with aggregation on a seasonal basis and the estimated effects of water quality on crop yield included. Other major assumptions incorporated in the model are listed below: 1) All demand sites are connected to the NWC, or can be connected with negligible fixed costs. 2) The new capacity which can be added at site i from marginal source j is known and fixed, having been selected separately from a small number of design alternatives. This capacity will be limited by pumping regulations in the case of SGW, population size at the demand site for TWW, and the probability distribution of rainfall for ROW. In addition, budgetary constraints at the site will limit the capacity which can be added. Existing capacities of marginal sources are assumed either to be negligible or operating at known volumes which can be netted from the total requirements at the site. 3) There are no left-over annual water inventories at the local sites. However, inventories can be stored from one season to the next in the same year. (See Rao et al. [7] on this issue.) 4) The quality requirement will vary among the different crops planted at a demand site, as well as during the life cycle of each crop. It is assumed that sufficient storage and handling facilities exist to meet these individual requirements, and that the costs associated with mixing water from different sources can be neglected. For purposes o f analysis, it is also assumed that the quality requirements at a site can be summarized by a net annual quality constraint. 5) The multi-quality water supply system will be described by a cost function to be minimized. Since the major components are variable, this cost will be conveniently expressed in terms of an equivalent annual amount (EAC i) at each site i. Capital investments at the sites are amortized assuming a fixed investment period for all new assests. Thus, the equivalent annual cost at site i is given by EACi =

(1)

~_,aijqij + "r ~_, bijyij Vi, j

j~4

where aii is the variable cost coefficient ( $ / m 3) for supplying demand site i from source j, qij is the volume supplied (m3/year) to demand site i from source j, z is the capital recovery factor computed for the fixed investment period and an accepted rate of return, bij is the capital investment ($) at demand site i for new capacity from marginal source j, Yij is an indicator v a r i a b l e (Yij = 1 if new capacity is added, 0 otherwise), and j = 1, 2, 3, 4 for SGW, ROW, T W W and NWC respectively. Most of the water consumed in the Negev is in the agricultural sector, to which we restrict our attention. Thus, the minimization of cost assumes that the crops selected and quantities planted have been determined beforehand, so that each site generates a fixed revenue. To proceed, the global optimization model takes the form of a mixed binary integer linear program given below for a region containing n demand sites.

The Regional Problem (RP): n

4

n

E i=1 j = l

aijqij -4- ~" E

Minimize Z = E subject to

3

E bijYij

i=1 j = l

(2)

J. Brimberget al. / European Journal of OperationalResearch 81 (1995)35-49

38

(i) Regional constraint {maximum supply from the NWC}:

(3)

~ qi4 <- S, i=1

(ii) Local constraints: 4 E qij =Di, j=l

Vi,

4 E Pijqij >--"YiDi, '¢i,

{demand}

(4)

{quality}

(5)

{fNed cost}

(6)

{capacity}

(7)

j=l

qij - MYij < O, Vi, j ~ 4, qiy <_Aiy,

Vi, j ~ 4,

(iii) Non-negativity and integrality constraints:

qij > O, Vi, j,

Yij = 0, 1,

Vi,

j 4: 4.

(8)

H e r e S denotes the maximum total volume (m3/year) supplied to the region by the NWC, which may be assumed to be constant under a short to medium range planning horizon, D i is the total annual demand (m3/year) at site i, Yi is the mean unit quality required at site i averaged over the year, Pij is the measured quality of the water at demand site i supplied from source j, M is a very large number, Aij is the new capacity (m3/year) added at site i from marginal source j ( j = 1, 2, 3) given that a decision to do so has been taken, and the remaining symbols are defined previously. The quality parameters Yi, Pij are expressed in terms of the negative of electrical conductivity (EC measured in units of deci Simmens per meter (dS/m)). In this way, larger algebraic values signify better quality (or less salt load). The quality of local sources such as SGW may vary considerably depending on the exact location of the source. Hence, in order not to lose generality, the parameter Pij depends on both the source j and the site i. Also recall that stationary conditions are assumed. Therefore, there is no advantage in having excess capacity (Z~ij -- q i ) carried over from one year to the next, since this simply incurs a variable cost before it is necessary. The fixed cost constraint ensures a capital investment b~j is incurred in the objective function if new capacity from marginal source j is added at site i. This follows from the fact that if qgj > O, the constraint must set Yij = 1. However, note that if qij = 0, the constraint will set y~j = 0, since otherwise a fixed cost is added unnecessarily to the objective function. The value assigned to M must be sufficiently large so that

M > max Aij , Vi,

j ~ 4.

(9)

3. Operational analysis at the local level

Let us consider a relaxed version of the local problem at demand site i where all three marginal sources are developed (Yij = 1, j = 1, 2, 3), and the supply from the NWC has been s e t (qi4 is fixed). Under these relaxed conditions at each site, the global problem decomposes into n local problems. The local problem at site i = 1. . . . . n takes the form of a simple linear program given by:

J. Brimberg et al./ European Journal of Operational Research 81 (1995) 35-49

39

Local Problem (LPI): 3

min

zi = ~ aijqiy j=l

s.t.

~ qij = di, j=l

(10)

3

(11)

3

~-~ Pijqiy >- Q i ,

(12)

j=l

O
j = 1,2,3,

(13)

where d i = O i - q i 4 and Qi = "YiOi --Piaqi4" In the subsequent analysis of (LP1), the index 'i' is omitted to simplify the notation. We first note the following important relation between the water qualities of the marginal sources: (14)

Pl
where once again j = 1, 2, 3 for SGW, ROW and TWW, respectively. This ordering reflects the fact that among the marginal sources, runoff water has the best quality, while saline groundwater has by far the worst.

3.1. Feasibility conditions A necessary condition for (LP1) to be feasible is that 3

E a; _>d,

(lS)

j=l

or that the total capacity exceeds total demand. To identify an optimal solution for (LP1) we consider the following two cases: 3 Case I.; Ey=lAy = d. This is the trMal case for which the unique solution q~ =/11, q2 = '42, q3 = A3, 3 3 z = E / _ ~ a j a / , is optimal if it satisfies the feasibility condition (12), i.e., ~,j_lpjAj > O. 3 Case lI: Ej=IAj > d > 0. This is a non-trivial case. (LP1) is now feasibleif, and only if, P2 min(a2, d) +P3 min(a3, d - m i n ( a 2, d))

+p, min(al, d - min(a3, d - min(A2, d)) - m i n ( A 2 , d ) ) > Q.

(16)

The feasibility condition stems from evaluating the optimal solution of the following linear programming problem to maximize the net quality from the three marginal sources in light of the precedence relation on water qualities in (14). 3

Max

~ &qy j=l

s.t.

~qj=d

and O
j=1,2,3.

Given that the above-mentioned feasibility condition is satisfied, then the standard form of (LP1), which consists of 5 constraints, 3 decision variables, 1 surplus and 3 slack variables, has an optimal solution. To locate it we first denote by u the surplus variable in the quality constraint, and by v~, v 2 and v3, the slack variables of qx, qa and q3 in their respective capacity constraints.

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40

3.2. Evaluation o f basic solutions

The basic solutions may be partitioned into the following subsets: (i) q l = £11, q2 = A2 and q3 = ':13; (ii) 0 < qj < As., j = 1, 2, 3; (iii) At least one of the q/s is equal to zero; (iv) Two of the q/s are positive and strictly less than their capacities, and the third is determined at its capacity; (v) One of the q/s is positive and strictly less than its capacity, and the other two are determined at their capacities. The elimination of basic solutions as candidates for optimality is justified by the following arguments: • A basic solution satisfying condition (i) is infeasible for case II. • A basic solution cannot satisfy condition (ii), since there can only be five variables in the basis. • A basic solution which satisfies condition (iii) may be ignored, since a subset of {1,2,3} yields the same solution at a lower equivalent annual cost. Thus, only basic solutions satisfying conditions (iv) or (v) should be evaluated. Lemmas 1 to 3 provide optimality conditions for basic solutions of these two types. (See Appendix A for proofs.) Lemma 1. Without loss o f generality, assume that ql = A1 and 0 < qs < Aj for j = 2, 3, in a basic feasible solution satisfying condition (iv). Then a necessary condition for this solution to be optimal is that a 3 < a 2. This result indicates that if an optimal solution exists with the aggregate quality of water equal to the minimal required level Q, then at this optimum two sources are developed with production rates below their capacities, where one developed source is inferior to the other in terms of its variable cost and preferred in terms of its quality. Lemma 2. Without loss o f generality, assume that ql < A1, q2 = A2 and q3 = A3, in a non-degenerate basic feasible solution satisfying condition (v). Then a necessary condition for this solution to be optimal is that a I _> max{a2, a3}. The preceding result holds for an optimal solution which meets the total demand at a quality level above the minimum. It is interesting to note that this result is independent of the p / s since the quality constraint is not binding. Also note that Lemmas 1 and 2 specify necessary conditions for optimality that are independent of the right-hand side parameters of (LP1). Lemma 3. Suppose (ql, A2' /ta3) is a feasible solution o f (LP1). Then a 1 >_max{a2, a 3} is a sufficient condition for this solution to be optimal. Furthermore, if the inequality is satisfied in a strict sense, this solution is the unique optimal. For qr < Ar and qj = Aj for all j v~ r, where r is a source which satisfies the condition a r >__maxj{aj}, the objective function of (LPI) is given by = E

- ar)aj + ard.

(17)

41

J. Brimberg et al./ European Journal of Operational Research 81 (1995) 35-49

Trivially, it is observed from (17) that (Oz/Od) > 0 for a basic solution satisfying Lemma 2. On the other hand, the objective function for a basic solution satisfying Lemma 1 is not necessarily monotonically increasing in d. As an example, consider (/12, q2, q3) for which

[a2-a3 P2 - - P 3

a2(Pl-P3) +a3(pz-P,)

~ P2 - - P 3

-1 a l ( P 3 - P 2 )

P2 - - P 3

At •

Recalling that

e = yD -P4q4

=

(T --P4) D +P4 d,

(18)

we obtain 0z --= Od

a3(P2 - P 4 ) - az(p3 - P 4 )

>0 --

P2 --P3

P2 - P 4 P3 - P 4 i f f - - > - a2 a3

3.3. Sensitivity analysis on d As a next step, it is important to consider a sensitivity analysis of (LP1) on the parameter d, the total supply at site i from the marginal sources there• Clearly, increasing d is equivalent to decreasing the supply to site i from the NWC. The sensitivity analysis permits us to study the effect this has on the optimal solution of the operational problem at the site. We proceed by first determining the feasibility conditions for basic solutions of the types covered by Lemmas 1 and 2. In the case of Lemma 2, suppose without loss of generality that a 1 > max{a2, a3}. Then (ql, /12,/13) is feasible and optimal provided that /

A2 "[-/13 ~
(19)

Upper and lower bounds on d are also readily derived to guarantee feasibility of solutions satisfying Lemma 1. For example, if we consider (A1, q2, q3), and combine the constraints 0 < q2-
(20a)

and

(P2 - P I ) / 1 a - (P4 - -/)D _< (P2 -p4)d <-(P2 -P3)/13 + (P2 -P~)/11 - (P4 - ~,)D. Thus, assuming for illustrative purposes that P2 >P4 >P3, it follows that max

(P4-T)D-(Pa-pl)A1 .... 174 -- P3

,

(P2"Pl)/1a-(P4-T) -----P2 -- P4

< d_< min( ( P 2 - P 3 ) A 2 + ( P 4 - y ) D P4 - P3

(20b)

D)

( P 3 - P ~ ) A1,

t

( p 2 - p 3 ) / 1 3 + (p2 - p ~ ) a l - ( p 4 - T)D ] P2 - P4

]

(21)

In similar fashion, upper and lower bounds on d can be derived for basic solutions (q~, /12, q3) and (ql, q2, A3), using the appropriate interchange of indices in (20a) and (20b). The details are omitted here.

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J. Brimberget al. / European Journal of Operational Research 81 (1995) 35-49

In linear programming theory, it is well known from right-hand side parametric analysis that the objective function z is a convex piecewise linear function of d. The results presented in the following theorem are more specific. (See Appendix A for a proof.)

Theorem 1. The objective function z in the relaxed problem (LP1) is a linear function o f d, or at most it consists o f two piecewise linear segments.

By computing feasible sets of d for each solution type using relations of the form (19) and (21), and eliminating non-optimal candidates by Lemmas 1 and 2, it becomes a straightforward procedure to compute the optimal solution as a function of d. By T h e o r e m 1, we know that the lower envelope of basic feasible solutions is a linear function of d, or in special cases non-linear with exactly two piecewise linear segments. 3.4. Optimal policy at a site

So far the operational problem considers the case where all sources are developed. Alternative relaxed LP problems consider only one or two marginal sources. A relaxed LP with two sources and the structure of (LP1) consists of 4 constraints and 5 variables in its standard form. For j = 1, 2, the basic solutions which are candidates for optimality are (ql, q2) and (/11, q2) if a 1 < a 2. The analysis of (LP1) is simplified for this problem and the piecewise linear structure of z with closed form solutions can be trivially obtained. Closed form solutions can be derived for the relaxed LP with a single source by inspection. For practical purposes, the objective function of the non-relaxed problem (the integer program) should be derived relative to the seven objective functions for the seven relaxed LP problems (associated with developing the following subsets of marginal sources: {1,2,3}, {1,2}, {2,3}, {1,3}, {1}, {2}, {3}) and include both fixed and variable costs. The variable cost of supplying water from the NWC (a4q 4 = a4(D - d)) must be added to z as well as the fixed capital recovery cost to obtain the total cost at the site. A lower envelope of the derived objective functions provides an optimal solution as a function of d, the total supply at the site from the local marginal sources.

4. The regional problem The relatively simple structure of the local problem allows the decision maker to focus on the conditions that lead to feasibility and optimality, and the types of optimal solutions and their ordering (Lemmas 1, 2 and 3, and T h e o r e m 1), for alternative settings of the problem. This feature of the local problem is useful for solving the regional problem (RP) of minimizing the total fixed and variable costs subject to an annual supply constraint on the amount of water provided by the NWC to the region. Since the regional problem is NP-complete, a heuristic algorithm will be developed to find near-optimal solutions in polynomial time. The heuristic uses information obtained on the breakpoints of the local problems to construct a Lagrangian relaxation. The details are given in Appendix B. It is obvious that at the optimum, at most one site needs to be evaluated at a point which is not a breakpoint of its total annual cost function. Thus, a total (NP) enumeration of the breakpoints for n sub-problems of type L R m in Appendix B will eventually yield an optimal solution to the regional problem. However, as outlined in Appendix B, a simple decomposition method may be used instead as a heuristic approach to solve L R m. The decomposition of the Lagrangian relaxation provides crucial information which relates the local and regional levels of decision-making. This information is contained in the Lagrange multiplier h which

J. Brirnberg et al. ~European Journal of Operational Research 81 (1995) 35-49

43

can be viewed as a marginal cost of replacing the regional supply by local sources. For example, consider a breakpoint value of A in the function ~br(A) for site r given by Cr,2 - Cr,1 hi

dr, 1 - dr, 2 "

The numerator (Cr, 2 - Cr, ~) gives the incremental cost in going from breakpoint 1 to breakpoint 2 in the total annual cost function for site r. Meanwhile, (d~, 1 - dr, 2) gives the decrease in supply from the NWC to site r. Thus, h 1 provides the marginal cost in moving from breakpoint 1 to breakpoint 2. By calculating the h/ at the various sites and arranging them in increasing order, we are able to sequence the breakpoints of the local cost functions in order of increasing marginal cost. This greedy approach enables us to obtain a good solution to the regional problem for an arbitrary supply (S) from the NWC, by choosing the cheapest marginal sources to make up the deficit while maintaining feasibility at the sites. Another advantage is that local and regional planners may now work together to implement marginal sources in a cost-efficient fashion while reducing the regional demand on the NWC in stages. We conclude this section by outlining the approximate solution procedure for RP ( t o be read in conjunction with Appendix B). Heuristic Algorithm

~r(/~) for r = 1. . . . . n, and arrange these values in increasing order. Denote the resulting sequence by F = {A1,..., AM}. Step 2. Repeat for m = 1 . . . . , n: (i) Remove from F the breakpoint values which belong uniquely to ~bm (i.e., site rn). (ii) E n u m e r a t e the remaining Ai ~ F in increasing order, calculating at each value the left-over supply qm,4 from the NWC to site rn. If qm,4 is less than the minimum amount required for a feasible solution at site rn, go to the next Ai. Otherwise, calculate the total regional cost, Vm(A)+ Zm(qm,4) + "~qm,4, of the current feasible solution. Terminate the enumeration the first time qm,4 ~>Dm (demand at site m) and the current feasible solution is inferior to the preceding one. (iii) Retain the best feasible solution found in the preceding enumeration. If none exists, stop. The regional problem (RP) is infeasible for the given supply S from the NWC. Step 3. Choose the best solution from the n candidates found in Step 2.

Step 1. Calculate the breakpoint values of

The algorithm requires O(n 3) computations. Due to its greedy nature, a near-optimal solution is expected if the number of local demand sites (n) is large. The solution also provides a bound, if further improvements are attempted using a branching algorithm.

5. The case of the Negev Desert

Six representative demand sites were chosen from the Negev region to form a pilot study. These sites are listed in Table 1 with their annual demands and water quality requirements in units of electrical conductivity ( d S / m ) . The annual quality requirement at a site is determined as a weighted average of the quality requirements of the individual crops planted there. Due to insufficient cost data, it was decided to delete runoff water from the study at this stage. As well, treated wastewater was not considered a viable option at sites with small population sizes. Those capacities which were included in the analysis are listed in Table 2 along with associated capital

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44

investments. All investments are amortized at a discount rate of 10% per year, and assuming perpetual asset life. Except for saline groundwater, the quality of the water supplied is assumed to be independent of the site location. For this study, the qualities from the N W C and from treated wastewater in units of electrical conductivity (EC) are conservatively taken as 0.3 d S / m in order to favour these sources relative to saline groundwater. T h e quality of saline groundwater is site dependent and may vary a lot depending on depth and location of the well (see Table 3). Prices charged by the N W C at each site are on a unit volume basis. The cost of pumping saline groundwater is based on an energy requirement of 0.00275 kwhr for each cubic m e t e r of water raised one m e t e r in elevation. Energy losses in the pipe are not included in this figure. Resulting cost coefficients for these two sources are given in Table 4. Annual maintenance and operation expenses for a wastewater treatment plant are charged at a rate of $ 0 . 1 1 / m 3 of effluent treated at any site. All other variable costs, such as on-site distribution, are assumed to be relatively small and therefore neglected. A local analysis was carried out at each site. Only one breakpoint value of A greater than zero was

Table 1 Demand and required quality at target sites i

Site

Demand (10 3 m 3 / y r )

Quality required (dS/m)

1 2 3 4 5 6

Central Arava Southern Arava Nitzana Beer-Sheva Revivim Arad

18 750 30 800 51000 25 200 15 000 40 000

2.16 2.43 2.24 3.09 1.73 2.50

Table 2 Capacities and investments Total capacity (103 m 3 / y e a r )

1 2 3 4 5 6

Present value ($) of capital investment

Saline groundwater

Treated wastewater

Saline groundwater

Treated wastewater

16500 21800 17 900 13 000 6 000 20 200

6 600 1800

534000 695 000 940 000 384 000 1603 000 1243 000

3 425 000 1390 000

Table 3 Estimated values for quality of saline groundwater i

1

2

3

4

5

6

EC ( d S / m )

1.9

4.2

7.4

1.8

6.0

3.1

J. Brimberg et aL / European Journal of Operational Research 81 (1995) 35-49

45

Table 4 Variable costs ( $ / m 3) i

National W a t e r Carrier

Saline groundwater a

1 2 3 4 5 6

0.25 0.35 0.24 0.18 0.15 0.17

0.07 0.09 0.12 0.05 0.22 0~15

a For utility rate of 10¢ per kWhr.

found, and this occurred at site 5 (Revivim). Thus we conclude, based on current prices charged by the NWC and the estimated data given above, that it is economical to invest immediately in all proposed new capacities of SGW and TWW except at site 5, irrespective of the limit on supply to the region from the NWC. The total volume available (S) from the NWC and the price of energy for pumping saline groundwater from deep wells were chosen next as critical parameters for a sensitivity analysis. The levels of supply investigated from the NWC were at 100%, 85%, 70% and 50% of the total annual requirements of the demand sites. Energy prices charged for pumpage were at 10¢, 20¢, 30¢ and 40¢ per kWhr, covering a broad range of possibilities. Fig. 1 shows the effect on total cost of the optimal solution of a reduction in supply from the NWC. As expected, the total cost increases (or remains the same) as the supply from the NWC decreases. However, it is interesting to note the insensitivity of the cost curves until S drops below 70% of the total demand. This signifies that saline groundwater is competitive with the NWC over a broad range of energy prices. Also note that at lower levels of supply from the NWC ( < 70%), the increase in total cost is much steeper at higher energy prices. This results from the necessity at lower levels of S to use groundwater sources which are uneconomical at high energy prices. For a detailed account of marginal sources at the individual sites, the interested reader is referred to Brimberg and Oron [1].

70 A 60

%

,

5O

40¢&w hr

30¢/kwhr

"~" ~ .

& 40

20¢/kw 1,--

.

hr

" " "

-

~

~ "

"

"[3

. . . . . . . . .

"{53

. . . . . . . . .

t.,)

3o

l O¢/ k w h r P-- . . . . .

20

i

50

.

I

60

.

.

.

i

.

.

X-. .

I

70

.

.

.

i

.

>6- . . . . . . .

I

80

i

I

90

Supply from NWC, S (% of total d e m a n d )

Fig. 1. Total cost of optima] solution.

K

1 O0

46

J. Brimberget al. /European Journal of OperationalResearch 81 (1995)35-49

6. C o n c l u s i o n s

An optimization model is presented to assist local and regional planners to integrate their plans for the economic development of marginal water sources in the Negev Desert in southern Israel. These localized sources of varying qualities are required in the short to medium range to supplement an overloaded regional supply by the National Water Carrier. The local problem is analyzed in depth to determine conditions for feasibility and optimality at this level. The simple structure of the local problem is used to formulate an efficient heuristic approach for obtaining near-optimal solutions to the regional problem. The methodology and concepts presented here may be generalized to water supply problems of a similar nature in other arid zones. Future research will deal with the extension of the analysis and the development of a decision support system to forecast runoff water and improve the reliability of the model's parameters. (See Dyer et al. [2] for similar systems developed for oil and gas exploration activities.)

Appendix A P r o o f of Lemma 1. The basic decision variables are given by ql, q2, q3, /)2 and u 3. Denote the

corresponding basis vectors by the columns of matrix B of order 5 × 5. Then for optimality the following two relations must be satisfied:

[02][] 112 q3

B qa

;

A1

=

U2 U3

;

¢:~ B-1 A1 =

(A.1)

ql

[a2J v2 A3 /33

A2 A3

and

-- [ a 2 , a3, a 1, 0, 0 ] B -1

0 -1 0 0 0

0 0 1 > [0, 0]. 0 0

(A.2)

Relation (A.1) is required for feasibility, while (A.2) gives the optimality condition on the reduced cost coefficients of u and v 1. The inverse of B is given by P3 Bml~

1 133 --P2

-1

Pl-P3

0

0

-P2

1

P2-Pl

0

0

0

0

P3-P2

0

0

-P3

1

P3-Pl

P3-P2

0

P2

--l

Pl--P2

0

P3--P2

J. Brimberget al. / European Journal of OperationalResearch 81 (1995) 35-49

47

and hence,

q2 = [ P3 d - Q + ( Pl - P 3 ) / 1 1 ] / ( P3 - P 2 ) ,

(A.3a)

q3 = [ - P 2 d + Q + ( p 2 - P l ) A a ] / ( P 3 - P 2 ) ,

(A.3b)

ql =/11,

(A.3c)

U2 =/12 -- q2 and v 3 =/13 - q3.

Relation (A.2) becomes 1 - [ a 2 , a3, a 1, O, 0]

[

Pl --P3 -

--1

P2 - - P l

0

P3 --P2

--1

P3 - - P l

1

Pl --P2

a2(Pa-P3) + aa(P2-Pl)

a~3 - a 3 -P2

+ al(P3-P2)

> [0, 0 I.

(A.4)

P3 --P2

Thus, relation (A.2) requires (a 3 --a2)/(p3--p2)_> 0. Since P3 < P 2 , we conclude that a necessary condition for the given solution to be optimal is that a 3 N a 2. [] Proof of L e m m a 2. T h e principles of the proof are similar to those of I_emma 1. The basic variables are now ql, u, vl, qa and q3. D e n o t e by A the corresponding matrix of basis vectors. It is easy to verify that its inverse is given by

A -1 =

1

0

0

-1

-1

ql

d

Pl -1

-1 0

0

P2-Pl

P3-Pl

u

Q

1

1

1

0

0

0

1

0

0

0

0

0

1

Iil

where

A

/31

=

/12

q2

/12

q3

A3

[-1-11

The optimality condition on the reduced cost coefficients of v 2 and v 3 is satisfied if

_ [ a l , 0 , O, aa, a3] h

or a I ~ max{a2, a3}.

1

= - - [ a l , 0, O, aa, a3]

/92 - - P l 1

P3 - - P l 1

1

0 ~

0

1

> [0, 0],

[]

Proof of L e m m a 3. T h e only other candidates which need to be considered for optimality are solutions of the type covered by L e m m a 1. D e n o t e these solutions by (/11, q2, q3), (ql,/12, q3) and (ql, q2, /13), and suppose that these solutions are feasible. It is easy to show that z(A 1, q2, q3) > z(ql, /12, /13), where z ( - ) is the objective function of (LP1) evaluated for a given solution. Since ql +/12 +/13 = A1 + q2 + q3 = d, then (/12 - q2) + (/13 - q3) = (/11 - ql)- Using a I > max{a 2, a3}, we obtain a2(A2 - q2) + a3(A3 -- q3) < a~(Aa -- ql)

48

Jr. Brimberg et al. / European Journal of Operational Research 81 (1995) 35-49

or a l q 1 + a2A 2 -t- a3A 3 ~< a l A 1 + a2q 2 -t- a3q 3.

Also by similar arguments it can be shown that z(q 1, A z, A 3) < Min{z(q 1, A2, q3), Z(ql, q2, A3)}" Furthermore, it is clear that the preceding relations will be satisfied strictly if a 1 > max{a2, a3}. [] Proof of Theorem 1. At a breakpoint where the slope (Oz/Od) changes, the following properties must hold: (i) two of the three sources are at capacity, and (ii) the quality constraint is binding. As d increases beyond the breakpoint value, the quality constraint may or may not remain binding. If the new basis is of the type satisfying Lemma 2, the quality constraint will have a surplus which increases with d. Thus, a subsequent breakpoint cannot exist. If the new basis is of the type satisfying Lemma 1, the quality constraint will remain binding as d increases, with the supply of the source not at capacity at the breakpoint increasing and the supply from one of the sources at capacity decreasing. The only possibilities are breakpoints of the type (A1, A2, q3) or (A 1, q2, '43), since Pl ( P j , Vj va 1. It is readily shown that a second breakpoint will result in the quality constraint being violated. Thus, we conclude that there can be at most two piecewise linear segments. []

Appendix B

Lagrangian relaxation of RP For each site m = 1 , . . . , n, construct the Lagrangian relaxation (LR m) given below, where every demand site except for the one denoted by rn is evaluated at the breakpoints of its total annual cost function. Wm(A)=min

~

~Ci,lYi,t+A

i=1

}~di,lYi,j-S

+zm(qm,4)

i = l l=1

i¢m

Li

s.t.

~Y/,l=l,

i=l,...,n, i~m,

l=1

A>0,

qm,4 > 0,

~ , t = 0, 1

Vi, l.

The indicator variables Yi,t, l = 1 , . . . , L i, are used to denote the breakpoint entering the solution at site i, i = 1. . . . . n (i ¢ m ) . Ci,t and di, l denote respectively the total annual cost at site i and quantity of water from the NWC consumed at site i associated with breakpoint l, Vi, l. Finally, qm,4 and Zm(qm,4)specify the left-over amount from the NWC allocated to site m, and the associated total annual cost of the optimal solution at site m, Neglecting the t e r m Zm(qm,4)in the Lagrangian function Wm(A) provides (n - 1) separable problems of the form

C~r(A)=min[l~=ltCr,l+Adr,l)Yr,l ] L~

s.t. E Y~,I= 1. I=1

J. Brimberg et al. /European Journal of Operational Research 81 (1995) 35-49

49

It is c l e a r t h a t ~br(h) is a concave, p i e c e w i s e l i n e a r f u n c t i o n o f A, w h i c h is easily c o n s t r u c t e d for t h e given v a l u e s o f t h e p a r a m e t e r s Cr, l, dr, t. T h e r e a r e at m o s t L~ b r e a k p o i n t v a l u e s o f I a s s o c i a t e d with 4Jr(t). F u r t h e r m o r e , as A m o v e s f r o m o n e b r e a k p o i n t v a l u e to t h e next higher, t h e q u a n t i t y o f w a t e r f r o m t h e N W C (dr, t) c o n s u m e d at site r d e c r e a s e s . Thus, if w e c o n s i d e r t h e c o n c a v e p i e c e w i s e l i n e a r function, n

Vm ( l~ ) :

E

f]~i (1~ ) - i~ s ,

i~l i~m

a f e a s i b l e s o l u t i o n to t h e r e g i o n a l p r o b l e m will b e f o u n d for sufficiently l a r g e A, such t h a t ~dilYil
p r o v i d e d t h a t a f e a s i b l e s o l u t i o n exists.

References [1] Brimberg, J., and Oron, G., "Development of marginal water sources: A case study", Ben-Gurion University of the Negev, The Institute for Desert Research, Kiryat Sdeboker, Israel, 1991. [2] Dyer, J.S., Lund, R.N., Larsen, J.B., Kumar, V., and Leone, R.P., "A decision support system for prioritizing oil and gas exploration activities", Operations Research 38 (1990) 386-396. [3] Fisher, M.L., "The Lagrangian relaxation method for solving integer programming problems", Management Science 27 (1981) 1-18. [4] Letey, J., and Dinar, A., "Simulated crop water production functions for several crops when irrigated with saline water", Hilgardia 54 (1986) 1-32. [5] Nativ, R., and Issar, A., "Problems of an over-developed water-system - The Israeli case", Water Quality Bulletin 13 (1988) 126-132. [6] Pasternak, D., Azoulai, A., Danon, A., Levi, S., DeMalach, Y., and Shalev, G., "Irrigation with brackish water under desert conditions, IV: Automated system to produce a range of salt concentrations in irrigation water for experimental plots", Agricultural Water Management 12 (1986) 137-147. [7] Rao, N.H., Sarm, P.B.S., and Chander, S., "Optimal multicrop allocation of seasonal and inter-seasonal irrigation water", Water Resources Research 26 (1990) 551-559. [8] Schwarz, J., Meidad, N., and Shamir, U., "Water quality management in regional system: Scientific basis for water resources management", in: Proceedings of the Jerusalem Symposium, IAHS Publication No. 153, September 1985, 341-349. [9] Willis, R., and Yeh, W.W-G., "Groundwater systems planning and management", Prentice-Hall, Englewood Cliffs, NJ, 1987. [10] Yeh, W.W-G., "Reservoir management and operations models: A state-of-the-art review", Water Resources Research 21 (1985) 1797-1818.