A mixed integer programming model for planning water resources investments is presented. The model is a sequencing model applied to the Vardar-Axios river basin in Yugoslavia and Greece. The structure of the model is outlined, and computational experience is described. The size of the model presented some difficulties, which are discussed along with the results to date. The experience with this model points to areas where further research is needed.Researchl supported (in part) by the U.S. Army Research Office (Durham) under Contract DAAG29-76-C-0064

Elliot, Dorothy P.

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-7, z;, ,.'- ':,' , '' - : - ' ' ' -w o r ' ,D a p.~e ~.. ~ ,.. : - ...--;:::,. . .E f f f ; At : :; : H 00L' A f .,','.' '  ' ,"-'''' ',' ", '' - ".' ' - , ', ' ''-' CHLJS:~ETTS INSIOOFECHOLOY-· -., J'·:·I ;r;l:--i 1.-i.-.Ii-.·' :'·I 1·....; :: i ··. --i '.. r:-F -·-1_.1 --.·..--·i-.· i-· i·.-i;'·.·'· ·:-::j'''1PLANNING INVESTMENTS IN WATER RESOURCES BYMIXED INTEGER PROGRAMMING:The Vardar-Axios River BasinbyDorothy P. ElliottOR 072-78 March 1978Researchl supported (in part) by the U.S. Army Research Office (Durham)under Contract DAAG29-76-C-0064.ABSTRACTA mixed integer programming model for planning water resourcesinvestments is presented. The model is a sequencing model applied tothe Vardar-Axios river basin in Yugoslavia and Greece. The structureof the model is outlined, and computational experience is described. Thesize of the model presented some difficulties, which are discussed alongwith the results to date. The experience with this model points to areaswhere further research is needed.I. IntroductionMany problems in the area of water resources can be modeled asmathematical programming problems. In particular, river basin planningcan be facilitated through the use of optimization techniques. Suchproblems may be solved using tools such as linear programming, networkoptimization, and mixed integer programming, depending on the nature ofthe problem. Numerous optimization models have been formulated and testedon specific problems, including single project models, where conflictinguses of the project must be balanced to produce the maximum benefitl,multiple project models, where flow along the river must be considered2,and entire basin models, where the needs of a large region must be takeninto account3. - Mixed integer programming in particular has been founduseful in problems involvingscheduling, sequencing, returns to scale,and plant location. One example of such a problem is a sequencing modelof the Vardar-Axios river basin (located in Yugoslavia and Greece). Thepurpose of this paper is to examine the problem and how it was solved,and to see how the techniques used can be generalized and applied to otherproblems.II. Statement of Model and DiscussionThe model is adapted from the one used by Cohon, Facet, Marks, andHaan4 in their work involving project selection and sequencing on theRio Colorado in Argentina. The project data is provided by a simulationmodel of the basin. Data required includes the locations, costs, benefits,and possible sizes of the projects, plus flow data for the river in itscurrent state. The task is to determine which projects should be builtand in which time period, and, in some cases, which of the proposedsizes. There are budget constraints, water availability limitations,and logical constraints. The objective is to maximize total benefits.The decision variables of the model are of two types; flow variablesand project variables. The river basin is marked with checkpoints where2flow data is available. (Many of these checkpoints are located at pro-posed sites of projects or immediately downstream.) The flow variables,which are continuous, give the flow at each checkpoint for each timeperiod for a given feasible sequence of projects. (Such a feasiblesequence of projects, representing a feasible solution to the integerpart of the problem, will be referred to hereafter as a configuration.)The project variables, which are zero-one variables, indicate whether ornot a given project is constructed during a given time period. The prob-lem allows for a planning horizon of more than one time period. Eachtime period is assumed to be long enough to allow for the completion ofprojects started at the beginning of the time period. The limitationon the number of periods in the planning horizon is that both the numberof columns and the number of rows grows as the number of time periods,drastically increasing the computational burden.The constraints fall into four classes: continuity, conditional,construction, and budget constraints. The continuity constraints balancethe flow from one checkpoint (or site) to the next to make sure thatthe configuration found by the model is consistent with the existing flowconditions in the river basin. These are equality constraints (the onlysuch constraints in the model) and there is one for each site in each timeperiod. The construction constraints simply ensure that a project is builtno more than once and in no more than one of the proposed sizes, if morethan one possible size is given by the data. There is one of these con-straints for each proposed project. The conditional constraints enforcelimitations such as precedence relationships (a reservoir must be presentfor a power plant to be built, for example). The budget constraintsguarantee that for each time period, total caplital costs for project con-struction will not exceed a given amount. There is one such constraintfor each time period. In the original version of the model there werealso population constraints to make sure that the number of people neededto develop the irrigation sites did not exceed the number of people avail-able. (These constraints were later deleted.)The objective function to be maximized is the sum of the benefits ofthe construction of a project minus the capital and operation and main-tenance costs of building that project. The benefit figures include3benefits from agriculture, power production, flood control, and recreationaluse, all taken from the data for the simulation model, from which the sizesand locations of the projects were also obtained.The detailed formulation follows:Original model: K SBudget: k bud projsi for all ik=l 8 si -proj SiPopulation: s Popsi IRRsi - Pi for all iTConstruction: proi 1 for all s,kConditional: PPVsi RESSJ for all s,iYSi - Zsi =0 for all s,iContinuity: (general form)D - D - RES + c REsi + E IRR F - Ffor all s,i(The form of continuity constraints depends on what projects areproposed at a site. At sites where water returns from an irrigation sitethere is also a (l-p)Es* term present, s* indicating location of irrigationsite outflow. In addition, each continuity row at a site where a branchflows in has a -1 in the column corresponding to the end site of theinflowing branch.)Objective function:max = PPPV PPV )EX EX P +[si -si )PPVsi i si EXRR _ d(IRR i + (- i)Yi RESDecision variables:Dsi release from reservoir at site s during period i, or flowat that point if reservoir has not been constructed.EXPsi construction or not of export to another basin from s during i4IRRsi construction or not of irrigation site at s during iPPVsi construction or not of power plant at s during iRES i construction or not of reservoir at s during iYsi construction or not of diversion(in) at s during iZsi construction or not of diversion(out) at s during iData variablesBi budget available in time period ic consumptive use of reservoir at s during construction andfillingbudki capital cost of project of type k at s constructed during iE diversion for irrigation at ssF streamflow measured at sSP i total number of persons available to develop irrigation areasin period iY yield from a reservoir at s<si value of capital and operation and maintenance costs fora project of type k constructed during i at ski value of time stream of benefits resulting from the construc-tion of a project of type k at s during i^/ irrigation loss coefficientThe model was applied to the Vardar-Axios river basin in two stages.Some revisions were required for the second stage. Population constraintswere dropped since it was believed that they would never become bindingin the river basin. Diversions, which in the first stage were treated astwo projects, one flowing into the basin and the other out, were mergedinto one. Also, some projects were proposed in several sizes so a newindex is added to the project decision variables. Thus the budget and con-struction constraint left-hand-sides are now also summed over z, as is theobjective function. Some conditional constraints were added to make certainirrigation sites dependent on some diversions. Also, a new set of projectswas added to represent municipal and industrial water supply projects, whichbehave in the model somewhat like irrigation sites since a fraction of thewater diverted is not returned to the river.For the first stage a simplified schematic of the basin and proposedprojects was used as the data for the model. This first stage includedfour time periods of five years each, for a planning horizon of 20 years.Seventy-one projects and 113 sites were included, yielding a matrix thatwas 608 x 736. The breakdown of projects was as follows:Reservoirs 18Irrigation sites 24Power plants 13Diversions (in) 6Diversions (out) 7Exports 371The second stage was more detailed and resulted in a significantlylarger problem. The number of proposed projects became 111. (Some ofthese had been combined into single projects in the first stage.) Inaddition, several sizes were proposed for a number of projects, resultingin not 4 but 8 or even 12 integer variables corresponding to some of theprojects. This resulted in an increase in the number of integer variablesfrom 284 to 628. The number of sites increased to 194. The size of thematrix became 1002 x 1404. The structure of the planning horizon wasnot changed.The water flow data chosen was for July, the driest month in thebasin, since if a configuration satisfied water limitations in that month,it definitely would during. the rest of the year. The breakdown of projectsin the second stage was:Reservoirs with power plants 10Reservoirs 20Power plants 2Diversions 8Irrigation sites 47Municipal & industrial supply 24111III. Computational Experience and ResultsThe model was set up using a matrix generating language and solvedusing an interactive branch-and-bound procedure on an IBM 370/168. Theinitial optimal linear programming solution had an objective function6value of 355.21 (in millions of dollars) as compared to the optimal integerprogramming solution of 217 (a gap of about 40%). This gap is largelyaccounted for by the fact that many irrigation sites were partially builtin the linear programming relaxation, contributing significantly to theobjective function, but had to be eliminated in the integer programmingsolution because if completely built they would be too large for the amountof water availble at the site. This suggested that perhaps alternatesizes should be considered, or that the data might be inaccurate, or both.A more detailed model was called for. Further evidence of this need wasthe fact that the integer programming solution included no reservoirsand no power plants. This called into question the benefit figures, thesizing of the projects, and some of the precedence relationships. Allthese considerations suggested a second pass at the river basin, usingless aggregate data than in the first stage.For the second stage more detailed and up-to-date information wasobtained. As noted above, changes in the formulation resulted in a largermodel, which was consequently much more difficult to handle.The optimal value for the linear programming relaxation was 29595.123millions of new dinars (Yugoslavian currency). This corresponds roughlyto 1500 million dollars. The large difference from the results of thefirst stage is due to revisions in benefits estimates as well as anincreased number of possible projects. Many projects in this solutionwere fractional, falling into two classes: some projects were to becompletely built, but over all four time periods, and others would be onlypartially constructed over the entire planning horizon. (This distinctionwas useful in planning the branch-and-bound strategy.) In this secondstage as in the first, the budget was more fully utilized toward the endof the planning horizon. This is not surprising since some expensiveprojects depended on the prior construction of relatively cheaper ones.The mixed integer program was attaked using branch-and-bound, andseveral attempts were made before a productive branching stategy wasfound. Simply finding a feasible integer solution (other than the trivialone of no project construction) turned out to be an enormous task. Thefirst goal was therefore to find a limited feasible solution: one in whichsplits in projects over time were allowed, so that the only variables7treated as. integer were the slack variables on the construction constraints,which were of the form xll + x 2 + x1 3 + x4 _ 1 . The branching strategyused in finding this solution was to force as many projects as possibleto be built until the water constraints are violated, and then force the marginalpne to zero and proceed. The intermediate solution found in this wayhad some projects built in stages over the planning horizon but all projectswere either completely built or not at all. The objective function valuefor this solution was 27274.269, about 8% less than the solution to thelinear programming relaxation. The budget was at its upper bound forthe third and fourth periods.The next stage in the solution would be to shift the projects aroundwithin the time periods, keeping the same solution to the intermediateproblem but looking at various feasible solutions stemming from that solution.Of course, there is no guarantee that the optimal solution to the wholeproblem is a descendant of that intermediate solution. It may well bea descesndant of another feasible solution to the intermediate problem.However, finding an integer solution as a descendant of that solutionwould at least provide a useful lower bound to the problem and indicatea range in which the optimal solution is to be found. Following sucha procedure with each of the feasible intermediate solutions would ultimatelyhave to lead to the optimal integer solution. (Unfortunately, this ismore easily said than done with a problem of this size. Storage require-ments and costs of the iterations needed in branching can become prohibitivelong before a solution is reached.)Once an integer solution is reached, much work remains to be done.Some sensitivity analysis is desirable, especially since budget figuresand cost and benefit estimates can be uncertain. This sensitivity analysisis a difficult problem in mixed integer programming. Furthermore, themodel (although large) left out some aspects that perhaps.should be in-cluded, such as power production requirements and agricultural water needs.The objective function also may be oversimplified, since some objectives,like power and recreational use, which were simply translated into dollarvalues, may be conflicting and involve tradeoffs. This would point to8the need for multiobjective analysis, another difficult area in mixedinteger programming.IV. Future ResearchThe experience gained in solving this problem indicates first thedifficulty of solving such a large-scale problem by branch-and-bound.The process is time-consuming and costly. Even using an interactivebranch-and-bound routine, which allowed choice of branches by onethoroughly familiar with the structure of the problem rather than simplyrandom branching, the problem was slow to solve. Furthermore, the storagerequirements were prodigious, even for the extremely sparse matrix inthis problem. For further work on this problem and others of this typeit seems that development of a decomposition strategy would be useful.But, more interestingly, it also seems that thorough treatment of theproblem in the areas of sensitivity analysis and multiple objectives willrequire some ground-breaking work in mixed integer programming techniques.Much research remains to be done here.References1. Thomas, Harold A., Jr. and Roger Revelle, "On the Efficient Use ofHigh Aswan Dam for Hydropower and Irrigation," Management Science, Vol.12, 1966, B296-B311.2. Roefs, Theodore G., and Lawrence D. Bodin, "Multireservoir Opera-tion Studies," Water Resources Research, Vol. 6, 1970, 410-420.3. Hufschmidt, Maynard F., "Field Level Planning of Water ResourceSystems," Water Resources Research, Vol. 1, 1965, 147-163.4. Cohon, J.L., T.B. Facet, A.H. Haan, and D.H. Marks, "MathematicalProgramming Models and Methodological Approaches for River Basin Plan-ning," Technical Report, Ralph M. Parsons Lab for Water Resources andHydrodynamics, M.I.T., Cambridge, Mass., 1973.5. Strzepek, Kenneth M., "Analysis of Multipurpose River Basin Plan-ning: Guidelines for Simulation Modelling," S.M. Thesis, Departmentof Civil Engineering, M.I.T., Cambridge, Mass., June 1977.

Planning Investments in Water Resources by Mixed-Integer Programming:  The Vardar-Axios River Basin

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Planning Investments in Water Resources by Mixed-Integer Programming: The Vardar-Axios River Basin

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