World Bank Reprint Series: Number 232 REP232 Joh4llnnes Bisschop, Wilfred Candler, John H. Duloy, and Gerald T. O'Mara The Indus Basiln Model: A Special Application of Two-Level Linear Programming Reprinted with permission from Mathematical Progrui:"ming Study, ?ol. 20 (1982), pp. 30-38. World Bank Reprints No. 200. J. B. Knight and R. H. Sabot, "The Ret 1uns to Education: Increaf.ing with Experience ar Decreasing with Expansion?" Oxford Bullelm of [con01'nlCS mId Statistics No. 20 1 . Mohan Munasinghe, "Optimal Electricity Supply. Reliability, Pricing, and System Planning/" Energy ECOn01tllf:' No. 202. Donald B. Keesing and Martin WoH, "Questions on International Trade in Textiles and Clothing," The World Economy No. 203. Peter T. Knight, "Brazilian Socioeconomic Development: Issue::. .,.H' the Eighties," World Development No. 204. Hollis B. Chenery, "Re'itructuring the World Economy: Round H," Forezgll A ffatr~ No. 205. Uma Lele and John W. tAelloL "Technological Change, Distributive and Labor Transfer in a Two ~.!ctor Economy," Oxford Erotlomic No 206. Gershon "Adoption of Interrelated Agricultural Innovations: Complementarity and tht> Impacts of Risk, Scale, and Credit," Ameri- {all Journal AgrIcultural [nmmmcs No. 207 Gt'r!-lhon Feder (:.Ind Gerald T. O'Mara, "Farm Si:~e and the Diffusion of Grt't:n Rl'volutil1n Technology," [«momic Dt'l,e[opmellf and Cultural , and IcOn Information anJ Innovation Diffusion: A Bayesian ApFroach, "/~mt'n(aJl Journal AgrIcultural [C0110111":5 No 208. Iv1ich,wl Anthropologists and D~velopme:1t- Orit'nh>d Rl'st~arch," lHdlgl>lWUS Anthropology 111 NOll-l,\'cster1l Countries No 209. n R. Lashman and Jt'remy Warford, "t [ealth World: Problems of Scarcity and Choice fllgltmd Journal of Medlcillt' No 210 Gt'orgp "Rt.iturns to Education: An Updated Interna- t ional " Comparatll'l' EductltlOll No. 211 K. and Alan Carroll, "The Spatia) Structure of Latin Amprican Citi,:~s, Jourllal of' Urhcm fC,HlW1lIfS No. 212. Hans P. Binswangt'r, "Income Distribution Effects of Technical Changt.>: Som(' Analytical " SoutlE East ASltW [C01l011l1C Rl'vlew No 213. Salah EI Capacity, the Demand for Revenue, and tht· " !OUYlwJ of [11l'rgll and Df'l dopmellt l No 214 de rv1.t.·lo and Sh{'rman "Trade Policy and Resource Allocation in tht.> PreSt'net.' of Product Diffprentiation," ReVIew of Fnm011l1CS al1d Statl-.Uc..;. No 215 Michat:"'i Cerni"a, "Modernization and f)('vt>lopment Potential of TraditIOnal Gra5s Roots Peasant Organizations," DirectIOns of Change: A1odcn;1::.aI101l Thcort/, Researdl, llnd Rcalitll's No. 210. Avishay Braverman and T. N. Srinivasan, "Credit and Sharecropping in Agrarian Societies," JO~'rnal of Dt'Z.'t1JOp1 :ent Economics No. 217 C('rl J. Dahlman and Larry E. Westphal, liThe rvieaning of Techno log i- cal Mastery in Relation to Transfer of Technology," Annals of the A mCY1can A(adt'mll of Puliflcal and SOCIal SoerIec Mathematical Programming Study 20 (1982) 30-J~ J'!urth-Holland Publishing Company THE INDUS BASIN MODEL: A SPECtAL AP'PLICATION OF TWO-LEVEL LINEAR PROGRAMMINf; Johannes BISSCHOP. Wilfred CANDLER. Jehn H. DULOY and Gerald T.O·MARA* Dtl:eiopmfnt Research Center. The World Bn.;.k. Washington. OC 20433. U.S.A. Received 24 January 1980 Revised manu\;:npt received ~6 May 1981 A ba~ic and verbal description nf the Indu ... Basin Model is presented. The model is an example of a "trategic planning exercl~e de ... igned to aid in the spt!cification of surface and ground water related policies in Paklst:.!.I. It is also a aptllication of the two-level linear programr.:"ing problem. The concert of luulti-lenl is introduced. and the general two-level linear program is de5>cl ibed as a nOli-convex problem. It is shown, however. that linear can be used i'l1 !he partil'ular appli ... ation of the Indus Basin Model Key word,~: Application, Multi-lt"vel Pro,~ran';:ling. Linc-ar Pmgramming. I. Introduction This paper is centered around an ongoing large-scale modeling effort designed to aid in the specification of and ground water policies in Pakistan. Rather than ,eiving the a df:tailed exposition of the mod~1 and some of its early (a task by itself). we have chosen to limit our model description to a minimum. and \~, ...)ncentrate instead on one aspect which, we feel, is often ignored or obscured in similar applications. This is the problem of hi~rar~hical declsion making f4. 5. 6]. Economic mod~ls designed f()r policy analysis usuaHy involve two kinds of agents: policy maker~ and policy receivers. If the policy receivers are optimizing agents, one is faced With a hierarchical decision making problem. or equivalently, a multi-level programming model. In the case of t1e Indus Basin Model. th~ government plays the role of the policy maker. while !he farmers play t"'e role of policy receiver~. government decides on surface water allocations, and sets taxes and/or subsidies. The farmers. in turn. react to the setting of these pp)icy instruments by uSing water (both surface and ground water) and choosing cropping patterns so as to maximize their own net Income. As th<: response surface of poiicy receivers is, in general not necessarily convex. an overall problem involving both types of agents may be a non-convex programming problem In that event, remedies such as using a weighted combinatIOn of the two objective func.tions will result in .. View!'. expressed are those of the authors and do not net.:e ...... anly reflect those of the World Bank or its affiliated organizations. 30 1. Bisschcp et al./lndus Basin Mndel 31 meaningless solutions. As ;.1 turns out, however, linear programming techniques can be used in the case of the Indus Basin problem. This ;~ discussed in Section 4. First, hOlvevcr, we win provide a verbal description of the Indus Basin Model in Section 2, followed by a generic introduction to the concept of multi-level programming in Sect;,on 3. The paper lonc1udes with the description of some numerk:al ~xperiments irL Section 5. 2. The Indus Basin Model Family Following the partition into the nations of India and Pakistan in 1947, India diverted for its own use some of the water from those rivers that formerly fed into parts of the irrigation system in what is now Pakistan. An international crisis emerged, which was finally resolved with the Indus Waters Treaty of 1960. Today. t~e \\ ater supply in Pakistan comes r.-aainly from the river Indus and its wlestermo~t tributarie~. Control of the water from the eastern tributary rivers is n:tained by India. The Treaty of 1960 resulted in replacement works consisting of ~wo larlEe reservoirs. three r~:!Jor barrages, 400 miles of new link canals for transferring water to aff(~cted areas. some remodeling of existing link canals and barrages, and a program of tubewells and drainage. Although these investments have all been madt'" , there is a continuous demand for further extensions and modification~ to the existing water supply system. The artificial redistribution of water is costly, and requires joint management of ground and surface y,ater. In many areas of the Indus B~l\in. for instance, tubewell development has not occurred, so that sustained application of canal water over dec2des has induced a contjnual rise in the water table. In some areas the water table is alreo:-dy quit.e near the surface. thereby creating problems of wattrlogging and salinity. Such condition" reduce the productivity of the land, or even take it out of production. On tt.~ other hand. too many tubewells in a region may r'!sult in a mining of the aqUifer. anl.i a pllsloIible influx of saline water into a sweet water aquifer. Even under the assumption that ground and loIurfa~~e water can be properly managed so as to avoid disastrous future consequences, there is still the basic problem of water al.l(lca!lon. Water form", the lifeblood of agriculture in Pak~ istan. and is a scarce resource. A~ the flow of water can be controlled and diverted at fllany poe'1ts in the surface water system, it is necessary to devise efficient wa.ter al1ocation scheme~ that will optimize some measlJre of regional welfare. This cannot be done without considering the use of water on individual farms. The Indus Basin Mudel Family was designed to relate agricultural development to the combined use and management of ground and surface water by incorporating these individual components into a unified mathematical framework. The basic structure of the Indus Basin Model can be visua1ized as fo]]ows. 1. Risschop et at/Indus Basin Moder The entire basin is partitioned into 53 irrigated regions, referred to as polygons. Each polygon is es:·entially homogeneous with respect to ground water quality, and preserves boundaries that ar\! significant to the ground water aquifer system. Lllf1Kllg(~S in water supply that arise from seepage of surface water to the aquifer and withdrawal of ground wat~r via tubewells or capillary action are explicitly modeled for each polygon, thereby interlocking the polygons. Each polygon also receives surf~ce water on a monthly basis fi'om one or more control points of the surface water delivery system. In addition to the above-mentioned water ~onstraints, e~ch ~olygonal mcdel has embedded in it a ~ingle farm level model to characterize the agricultural production ~ystem ..)f the are,\.. Such a farm level m,-,del simulates the resource allo;;ation of a sing~e representative farmer which determine the and disposition of 11 crops and 4 livestock commodities. Exogenous resource are imposed on land, labor aud canal water. The water and demand constraint of each farm level model tncludes estimates of ~l~"III'lIR>I'" from rainfall, evaporranspiration from the aquifer. the exogenous tubewell water. and the endogenous supp}} of when used to evaluate water allocation poli- are sometimes endogenous. There is, of course, the availability of surface water, and estimates averages and worst cases. The uncertainty associated ,U»lount of rainfall i~ not ~o important as its contribution to the overall and capital markets in Pakistan, it is assumed between polygons. This implies that the the underground flows between them. provinces or agro-c!imatic economic linkages polygons level in the hierarchy between ~implification is the can be ,justified i'1 the case of and cash crops on the are jointly "" odd pnce~ and product~ and are more of a .... r.'\hl~n1t price,,; llutside the ,cope of I"oftware. entire Indu ... Ba..,m Model is a linear more than :20000 the capability of eXIsting software for on that a simplification would be the water tahle in each polygon to a policy instn,lment. "tructural could be made such that the model contain', le"s than 000 constraint" \\ hich is solvable u..,ing a machine and arc. above introduction has been bnef and has .... n""...:,,1 many J. Bisschop et al.I Indus Basin Model 33 details. It aiso has made no mention of .~he extensive data gathering efforts that have taken place partially in support of the model. Despite this brevity and the lack of any mathematical equations (a detailed model statement is almost 40 pages), ~he reader must have gotten some impression of the structure and complexities that are captured by the system. With such a model. it is possible, among other things. to evaluate the effect of a wide range of water·related investment project~ ~tarting with dams and going aU the way down to improve- ments on wat<~r courses in the field. As individual f(·,rmers do not recognize their individual effects on ground water equilibrium which must be maintained over the long-run. the government must take into account the long4erm consequences of any water allocation scheme and the impact vf water related investments on equiiibrium. This expresses precisely the two-level aspect of the Indus Basin Model where some cO!lstraints are not formally recognized by the farmers (the policy receivers) even though the government (the policy maker) requires that they be satisfied. How the government might accomplish this is expJained in Section 4. First. however. we would like to introduce some basic notions on multi-Jevel pro~ gramming as most readers may not be familiar with this topic. 3. An introdu(~tion to muiti..lt'vel programming Vlhenever the~e is a set of nested optimization one cau of multi-level program This terminology was introduced by Candler and and several economic applications are di~cus~ed in their paper. A genl~ral mathematical definition multi-level programming is not attempted in this paper as it an unwieldy expression symbols and W'e I.:ho~en to a gen~ral definition of the pro- ... .,,, .. ,'u problem. leaving any generalizations to the reader. Assume that there is one maker (or 'outer' decision ,and one group of independent policy (the 'inner' decision Let x be the (11~ 1)-vector variables controlled by the outer decision while the I I)-vector), contains the decision variables of the inner decision maker. Then general programming problem can be written a~ Min f:( x. y), s.t. R~(X. y) =0 'Y) ?: O. 0.1) S.t. gl(X, y) = 0., hj(x. y) 2: O}. Note that the inner optimization problem is nothing else but one of the constraints of the outer problem. In the inner optimization problem itself the x 34 1. Bisst'hop et at/Indus Basin Model variables are considered as given. The outer objective function could be vacu- ous. but if the inner one is not, one is still faced with a two-level programming problem. The above problem can be viewed as a two-player Stackelberg game o!mpioying the leader-follower solution concept. The leader is assumed to have complete knowledge of the behavior of the follower. It is precisely this authority between !hl! leader and the follower that induces a natural solution to the above mathematic;al problem. provided, of course. that the optimal reaction y*(x) is unique for the relevant choices of x. If this is not the case, the above problem does not have a solution. and additional information must be added to the prob'em. Whenever y*(x) is not convex, the overall problem has local optimal solution~. As one cannot assume a priori that y*(x) is convex {even in the case of linear problems !). one must assume in general that the two-Jevel programming problem is non ..convex. This observation provides chaHenges to the mathemati- cal prDgramming community, but despair to an practitioners faced with multi- level programming problems and desiring globally optimal solutions. As we shall see next, however. there i~ a lar~e class of linear two-level programming problem~ that can be solved for global solutions using mixed integer-linear programming. Using the same (x. y)-notation as before. consider the following two-level proaramming problem. Min c J,t. b:-. x O. {min ely y Qy +- x ~ C 'yo is «I ",). Q i~ «" (' ,I ~,ymmetric. C hI ("~ x ",). and A" is j «nil "1) for i 1. 2 and j at the largest class programming that i~ still commercial flt,oftware. Consider the derivation. If one writes the Kuhn- Tucker conditions the inner optimization problem with x constant. one obtains a complementarity Embedding this linear com~ plementarity in the outer ll~'tlmization problem us problem 0.3). Mm c .~. Ii Lo x (ii) S, ~\. y. A. IL. W ? o. (iii) =0. i=1.2 ..... ml. IL,Y, :=. O. 1.2 .... "I' J. Bisschop et al./lndus Basin Model 35 In the above augment~d problem, only the constraints under (iii) are nonlinear. They can be replaced with SCI-called 'special ordered sets of type 1'. where at most one of the variables in the set can be positive. Replacing the constraints under (iii) with m, +"1 special ordered sets of this type. a mixed integer linear programming code such as APEX of CDC (which accepts continuous variables as part of its special ordered sets) can be Llsed to solve the above problem [1]. The proposition of using a mixed integer linear programming (:ode to solve linear two-level programming problems is an expensive one, as the&e codes are usually slow in finding a proven global optimal solution. It is certainly an unrealistic suggestion in the case of the Indus Basin Model where the augmented problem becomes too large for the machine. As it turns out. the Indus Masin Model contains specific characteristics that allows one to use linear program- ming techniques to solve the two-level programming problem. 4. The Indus Ba.,in Mod~1 as a special case of multi·~):vel programming In the Indus Basin Model each polygonal representative fa.'m is an in- dependent unit, so that summing the Individual farm-level objechve functions over all ~3 polygons provides us with a ~ingle objective function rep the entire agricult~Jral sector. Maximizing the aggregated net farm inl:ome can be considered as a proxy for maximizing welfare. In that sense both the govern- ment and the the same objectives. and no special distmction needs to be made. two groups do differ. howevc=r. in that the government wants to satisfy a set of political constraints and long-term ground water balance requirements that are the realm of the farmers. and, as such, are not recognized by them. Basin Model can therefore be viewed as a special example of a two-level programming problem where the outer objective function is vacuous. Using a notation similar to that of problem (3.2). we ,an write the following mathematical statement. x ~o. {Minimize (4.1 ) \il" Jl s.t. y ~ O}. Here both x and d are considered policy variables that pertain to the government. The vector x contains aJl variables that are not under the direct control of the farmers. and that are not in d. Surface water allocations to the farmers. for instance. comprise some of the x-components. The vector d represents a set of subsidies and taxes that the government wants to impose on farmers' water related activities. Note that these control variabJes enter the inner objective function in a multiplicative fashion, thereby making the above problem J. Bi3schop et at I Indu.'i Basin Model bilinear. A setting of both x and d wiH result in a response y* by the farmers which in conjunction with i:he x values mayor may not satisfy the outer constraints of (4.0. In order to solve l~he abo'Je problem (4.1), the following algorithm can be applied. Add for the moment the outer constraints to the inner optimizatiun problem. and set the vector d equal to zero. Next solve the resulting linear program, and denote the optimal solution as (x · .9). Let the vector w be the corresponding set of optimal dual prices associated with the outer constraints. Then if the government st~ts x = x and d T = ilT = - WT A2h one can verify, using duality theory. that y* = y is the optimal response for the farmers in the problem Minimize (c T+ ilT)y, ~lu.J) s.t. (4.2) By construction" the resulting pair (x. y*) = (x,.9) satisfies the outer con- straints. The intuitive reason as to why y* = y solves problem (4.2) is that the transformation of the inner objective function from c Ty to (cT + ijT)y has caused the outer constraints to become redundant as far as the farmers are concerned. Another way to express the same intuitive idea is to say tiT at if the outer constraints were added to problem (4.2), their shadow prices would be zero. It is important to note that after one computes the values of (x, il, y) for the augmented linear program, one must also solve the inner problem (4.2) with the policy variables fixed at ill. This h an unfortunate but necessary step to make sure that the optimal response y*(i, ill is unique. As we noted in Section 3, without this uniqueness property. the solution to (4.1) is not defined. In this case the inner agent has no a priori reason to choose the value y that also satisfies the outer constraints. A transfer payment (bribe) from the outer agent is needed to induce the inner agent to select the y that satisfies the outer constraints. Even though the vector y. ill odtained by the above procedure solves the two-level programming probJem (4.1). the actual values of .i and il may not be politicaUy acceptable. An example could the surface water allocation scheme. As each representative polygonal farmer has an equal weight in the aggregated farmers' objec~ive function. allocations of scarce water will naturany favor the efficient farmers. This could result ia optimal water allocations .i that essentialJy ignore c,ertain regions within the country. Such water allocations will be unac~ ceptable to the regionai government represe;ntatives who will insist that a minimurr percentage of surface water must be allocated to them. This Implies an increase in the number of outer constraints. which in turn requires the genera~ tion of a new solution y. ill. By comparing the optimal solutions and the corresponding objective function values. one can evaluate the impact such political constraints. Assuming that enough ~urface water allocation constraints have been added to the outer problem so as to render politically acceptable x -value~, the resulting J. Bisschop et at/Indus Basin Model 37 solution vector d in the two-level problem may contain excessive taxes and/or subsidies. One approach is to design a set of acceptable tax/subsidy programs in the form of the vectors ,.\, d 2, ··· , die. Then one can employ parametric pro- gramming on the inner objective function, and introduce subsequentially the tax packages d i up to some specific level Aidi where Aj are scalars. If these packages are well designed, they will tend to lead toward approximate sotations of the outer constraints. Any violation of these constraints can then be corrected using appropriate government investment programs in tubewells. drainage projects and/or surface water related projects. 5. Some computational results The Indus Basin study has not been completed at the time of this writing, and detailed results cannot be placed in the public domain. Nevertheless, we would like to report on three experiments (scenarios) involving the two-level for- mulation of the Indus Basin problem. The results of these scenarios are compared to the ~base C(hf', which assumes that surface water allocations are fixed on the basis of a 5-year historical average, and tt.at no ground water levels are to be enforced. In scenario A. it is assumed that surface water allocations are fixed on the basis of a 5-year historical average (just as in the base case), and that the 1977 -1978 levels of ground water are to be maintained. In Scenario R, it is assumed that a significant portion of surface water is allocated Oil the basis of existing historical water rights. and that the remaining portion is alll\)cated freely by the model, while stm requiring t.h~>t the 1977-1978 ground watef levels be maintained. Ih scenario C, it is a~;,umed that all surface water is allocated freely by the model, and that the 1977 income levels of all the polygons are to be guaran!eed, together with the requirer.lent that the 1977-1978 ground water levels are maintained. As these scenarios allow for increased freedom in th~ allocation of surface water, one would expect increases in agricultural production, and reductions in the: levels of taxes and subsidies on water. These expectations are supported by the f!'sults of the experiments. In scenario A, en ~orcing the ground wate~ balance requires taxes up to 250 rupees (S25'()() per acre foot and subsidieS oi ~p to llQ rupees ($11.00) per acre foot for some polygons. Agricultural valu~ added in domestic prices is decreased relative to the base scenario by approximately 1.5 billion rupees, which corresponds to a drop of roughly 5%. In scenario B, enforcing the ground water balance requires taxes of up to 125 rupees per acre foot. and subsidies of up to 105 rupees, while the corresponding taxes and subsidies in scenario C are up to 60 and 40 nlpees, respectively. For scenario B, the increase in agricultural value added relative to the base case is 2.8 billion rupees, which correspond!3 to an addition of roughly 10%, while the correspond· 38 J. Bisschop et al.IIndus Bas;n Model ing increase for scenario C is 4.3 ...mion rupees, or roughly 15%. Although one cannot make any definite conclusions, these experiments do point at the relath,~ importance of flexible surface water aHocations If ground water levels are to be maintained over long periods of time. 6. Summary and concludon In this paper, we h2.~'e attempted to provide the reader with some insight~ into the complexities of an on~oing '''1Qdeling exerc:se in a strate&ic planning environment. The main empnas;s, ho\\ ~ver, has been on the multi-level aspects of the problem, expiain!:ie '1e roles of b(·th the governmehi and the farmers. As multi-level programming i_1 not a widely ~ flown area within the field of mathe- matical programming, we ~]a'/e if.c1uded an introJl1ctiun to this important applied modeHng tool. Although linear programming techniques are usually not applic- able to the non-convex multi-level programming problem, it is shown how the special case of the Indus Basin problem forms an exception. 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