LINEAR PROGRAMMING APPLICATIONS IN WATER RESOURCES1

Applications of linear programming to water quality and water quantity problems are discussed, and a fairly comprehensive sample of recent literature in these areas is reviewed. Basic elements of linear programming are also discussed. Emphasis is placed on the elements of linear programming that make it a useful tool for analyzing water resource problems and the basic features of various water resource problems that render them amenable to meaningful analysis by linear programming.     

SOLID SET IRRIGATION SYSTEM DESIGN USING LINEAR PROGRAMMING1

: Irrigation systems can be formulated to enable linear programming to be used to obtain optimal systems. Linear programming is a readily available technique in most scientific and technical centers which have computer facilities. The objective function towards a minimal cost is based on a suggested classification of the system components, and the constraints mainly refer to required pressures and discharges. A case study in which linear programming is used to determine optimal pipe diameters is given for the solution of a solid set under tree sprinkler system.     

FLOOD CONTROL RESERVOIR OPERATIONS UNDER ENVIRONMENTAL RESTRAINTS1

ABSTRACT: A flood control reservoir protects valuable developments on the downstream flood plain by storing flood waters and releasing them at a rate that will reduce the downstream damage. The water surface level of the flood pool behind the dam can fluctuate considerably during the occurrence of a large magnitude flood causing the inundation of trees, low vegetation, and water based recreation facilities located in those areas of the flood pool area that are normally well above the water level. The amount of damage that will occur in the upper levels of the flood storage area will depend on the depth and duration of the inundation that occurs. This, in turn, is directly related to the operating policy for the reservoir. A dynamic programming optimization model of flood control reservoir operation is presented. This model determines the reservoir operating schedule that minimizes downstream flood damages. Various constraints are added to the model to account for the environmental impacts of long periods of flood storage.     

COMPARISON OF RESERVOIR LINEAR OPERATION RULES USING LINEAR AND DYNAMIC PROGRAMMING1

ABSTRACT: Mathematical optimization techniques are used to study the operation and design of a single, multi-purpose reservoir system. Optimal monthly release policies are derived for Hoover Reservoir, located in Central Ohio, using chance-constrained linear programming and dynamic programming-regression methodologies. Important characteristics of the former approach are derived, discussed, and graphically illustrated using Hoover Reservoir as a case example. Simulation procedures are used to examine and compare the overall performance of the optimal monthly reservoir release policies derived under the two approaches. Results indicate that, for the mean detention time and the corresponding safe yield target water supply release under existing design of Hoover Reservoir, the dynamic programming policies produce lower average annual losses (as defined by a two-sided quadratic loss function) while achieving at least as high reliability levels when compared to policies derived under the chance-constrained linear programming method. In making this comparison, the reservoir release policies, although not identical, are assumed to be linear. This restricted form of the release policy is necessary to make the chance-constrained programming method mathematically tractable.     

OPTIMIZING FLOOD CONTROL ALLOCATION FOR A MULTIPURPOSE RESERVOIR1

ABSTRACT. In the last decade much research has been devoted to applying the systems analysis approach to water resources problems. A popular research goal has been determination of the “best” method of operating a multipurpose reservoir. The goal of this study was to derive the economically optimum flood control diagram for a multipurpose reservoir by systems analysis. The technique employed to optimize the flood control diagram was programmed so that the optimization process could be applied to other multipurpose reservoirs. Two computer programs developed at the U.S. Army Corps of Engineers' Hydrologic Engineering Center were utilized with modifications to simulate the operation of Folsom Reservoir in central California. Economic analyses were incorporated along with an optimization technique into the reservoir operations program; and the resultant program was capable of routing a sequence of monthly reservoir inflows, computing benefits for various flood control diagrams (as dictated by the optimization procedure), and selecting the economically optimum flood control diagram. The univariate gradient technique was the optimization procedure employed. The two computer programs are on file at the Hydrologic Engineering Center in Davis, California.     

AN APPLICATION OF MATHEMATICAL PROGRAMMING IN PLANNING SURFACE WATER STORAGE1

Designing a surface reservoir involves the concept of reservoir yield. This concept embodies three basic information items: hydrologic regime, active storage volume, and reservoir release policy. In the actual case presented below, the magnitude of the active storage was prescribed by a legal procedure, so that the planning issue became that of determining the reservoir yield given the hydrological information. A stochastic dynamic programming model was formulated to derive a schedule of seasonal optimal reservoir releases and their respective probabilities of occurrence. This schedule is the reservoir yield. The yearly cycle was divided into three seasons representing the actual climatic conditions, and conditional probabilities linking streamflows in consecutive seasons were estimated. An operating policy was postulated, based on the same set of legal decisions that prescribed the active storage volume, and target reservoir releases were assumed. Similarly, target storages at the end of each season were set up. The optimizing/ minimizing criterion in the dynamic programming formulation was the sum of squares of deviations of actual releases and final storage volumes from their respective targets.     

THE RED RIVER FLOOD CONTROL SYSTEM AND RECENT FLOOD EVENTS1

ABSTRACT: The 1950 flood disaster in the Red River Valley, Manitoba, and particularly in Winnipeg made all levels of government aware of the need for control measures. The principal elements of the system which was implemented were two large excavated diversion channels, a storage reservoir, and ring dykes around several small communities. In terms of cost and size, the flood control system is the largest in Canada and despite Federal contributions amounting to nearly 60 percent of the final cost, it represented a considerable fiscal burden for the comparatively small population of Manitoba. Between the opening of the Red River Floodway in 1968 and 1979, a series of exceptional spring peak flows on the Red and Assiniboine Rivers demonstrated the benefits of such a system to a degree which could not have been anticipated at the time the projects were being considered. Furthermore, maximum spring discharges from 1913 to 1978 show a clear rising trend, indicating that the flood hazard is becoming even more severe than was initially assumed; if this trend continues, future benefits will continue to exceed expectations. The overall effectiveness of the hazard reduction program in the Red River Valley, however, has suffered from continued development in unprotected areas. Recent federal-provincial agreements have been reached which will substantially reduce this problem and place greater emphasis on improving the non-structural components of an overall flood hazard reduction program.     

APPLICATION OF LINEAR SYSTEM'S THEORY AND LINEAR PROGRAMMING TO GROUND WATER MANAGEMENT IN KANSAS1

ABSTRACT: A ground water management model based on the linear systems theory and the use of linear programming is formulated and solved. The model maximizes the total amount of pound water that can be pumped from the system subject to the physical capability of the system and institutional constraints. The results are compared With analytical and numerical solutions. Then, this model is applied to the Pawnee Valley area of south-central Kansas. The results of this application support the previous studies about the future ground water resources of the Valley. These results provide a guide for the ground water resources management of the area over the next ten years.     

A SEPARABLE LINEAR ALGORITHM FOR HYDROPOWER OPTIMIZATION1

ABSTRACT: A deterministic, separable, linear algorithm is presented for maximizing aggregate hydropower production. The method is iterative and amenable to solution using standard LP software. The utility of the technique is demonstrated using several test applications involving a hypothetical single-purpose hydropower reservoir and a monthly increment 20-year flow record from the Gunpowder River in Maryland. The separable linearized forms solved quickly using MPSX on a variety of IBM hardware: 3090-400 VF, 3084 QX, dual processor 4381-3, and an AT/370 personal computer. For comparison purposes, the original nonlinear nonseparable version of the model was also solved using MINOS. This yielded a value of aggregate hydropower marginally higher than that using MPSX. The separable, linearized methodology proved to be a useful and an efficient means of generating good starting points for MINOS. The use of these warm starts effected substantial reductions in MINOS execution times.     

A RISK-BASED APPROACH FOR FLOOD CONTROL OPERATION OF A MULTIPURPOSE RESERVOIR1

ABSTRACT: Many approaches are available for operation of a multipurpose reservoir during flood season; one of them is allocation of storage space for flood control. A methodology to determine a reservoir operation policy based on explicit risk consideration is presented. The objective of the formulation is to maximize the reservoir storage at the end of a flood season while ensuring that the risk of an overflow is within acceptable limits. The Dynamic Programming technique has been used to solve the problem. This approach has been applied to develop operation policies for an existing reservoir. The performance of the policy was evaluated through simulation and was found to be satisfactory.     

FLOOD PLAIN MODELING1

Computer simulation of river basin hydrology has rapidly progressed from an interesting academic exercise to a practical engineering procedure of increasing utility. Mathematical models of the many interrelated processes which occur in a basin have been developed along with efficient numerical procedures for their solution. The present paper is concerned with a particular model which has been used to describe the transformation of a temporally and spatially varying rainfall into a time history of stage and discharge on a flood plain. Although developed principally as a model of the physical processes involved, it is envisioned that the model can serve as one component of an information system for flood plain planning and management. The model consists of the following elements: (i) a rainfall simulation which generates stochastic inputs to the model according to specified rainfall statistics, (ii) a catchment-runoff model which converts the rainfall to surface runoff, (iii) a flood stage model which converts the surface runoff to time histories of flood stage and discharge. The model has been used to investigate the effect of various structural flood control measures in a basin and, in particular, to establish frequency-stage information for each of these. Of particular interest in development of the model have been recurring and partially unanswered questions relative to the proper balance among availability of data for use in the model, data requirements of the model and the objectives of the outputs produced by the model.     

SOLUTION OF A WATER RESOURCES PROBLEM WITH ECONOMIES OF SCALE BY LINEAR PROGRAMMING1

ABSTRACT Both because of its effectiveness and ease in use, linear programming has become progressively popular in water resources planning problems. Yet, the assumptions of linear construction costs can be misleading. Diseconomies of scale in construction can be handled by successive approximations to the cost function but problems with economies of scale yield paradoxical results when piecewise approximations are used. If significant economies of scale exist in only one facility, the solution to problems of this nature can be found using normal linear programming codes by successively adjusting the unit construction cost on that single facility to iteratively work toward the true optimal solution.     

LUMP SUM TAX LOSS REIMBURSEMENT UNDER THE THAMES RIVER FLOOD CONTROL COMPACT1

ABSTRACT: Under the terms of the Thames River Valley Flood Control Compact, Connecticut has been paying Massachusetts 40 percent of the annual property tax losses suffered by seven Massachusetts towns where four flood control structures were located. The permissable alternative of a single lump sum payment was investigated in the study summarized here. The lump sum payment should be the proportion of total benefits (flood control and recreation) from the four structures which Connecticut receives, multiplied by the present value of projected tax losses in the seven tom Flood control benefits and their distribution between the two states were fixed in the Compact, but a survey was necessary to determine recreational benefits and their distribution. Regression analysis of 1957 to 1978 tax loss data provided equations used to project future tax losses. Resent values of projected tax low were calculated using discount rates ranging from 6 to 12 percent. A plausible range of lump sum reinbursements as of 1979 was identified.     

MULTICRITERION DECISION MAKING FOR FLOOD CONTROL OPERATIONS: THEORY AND APPLICATIONS1

ABSTRACT: Flood control operations are generally complex in nature, as they are often associated with a large quantity of uncertain factors in combination of noncommensurable criteria. It has been found that the use of fuzzy set theory coupled with multicriterion decision making (MCDM) methods is very promising. In this paper, a multicriterion fuzzy recognition model is established for flood control operations. A subjective preference and iterative weights method is proposed for weight assessment. First, an initial solution of criteria weights, which is directly analyzed from the selected alternatives, is obtained by using the proposed fuzzy recognition model. Then, according to their knowledge related to real time flood operations, operators may modify the initial weights if necessary. When the relative membership grades of alternatives belonging to all rankings are fixed, a defuzzification equation is used to calculate the rank feature value vector. Based on the rank feature value vector, one can evaluate the alternative set. Two flood operation cases are provided to illustrate the application of the proposed model. With the incorporation of the operator's knowledge related to flood operations, the proposed model is flexible in giving weights and realizes more practical flood control operations.     

ECONOMIC ANALYSIS OF IRRIGATION PRODUCTION FUNCTIONS: AN APPLICATION OF LINEAR PROGRAMMING1

ABSTRACT: Economic analysis of irrigation production functions is discussed using linear programming. The method provides advantages over the partial one-crop type of analyses because it captures intercrop tradeoffs in water and land use in response to economic policies or changes in water supply. A numerical example is used.     

MODELING A SOCIOLOGICAL-HYDROLOGIC FLOOD CONTROL DECISION SYSTEM1

ABSTRACT: This work was the development of a model for analyzing the social components of a flood control or sociological-hydrologic decision process. A general conceptual system was developed from the study of an actual decision. Mathematical values were determined for the social and behavioral variables and these elements were transposed into a mathematical linear model providing a set of equations from which the system could be simulated with the computer.     

THE MEASUREMENT OF IMPACTS IN SMALL WATERSHEDS FOR FLOOD CONTROL1

ABSTRACT: An analogue method of ex post evaluation is proposed as a method of measuring the effectiveness of small watershed projects in obtaining flood control and economic benefits. Two watersheds were compared on a “with vs. without” project basis by both direct and indirect measurement of economic benefits. Direct measurements indicated that small watershed flood control projects did not generate the expected economic benefits. However, the indirect measurements of the same watersheds using land value enhancement as a surrogate suggested that the expected economic benefits were reflected in differential land values. The economic efficiency of small watershed projects should be measured ex ante and ex post on a “with vs. without” project basis rather than on a “before vs. after” basis whether using direct or surrogate variables.     

AN ARCHETYPAL LINEAR PROGRAMMING MODEL FOR OPTIMAL INTERTEMPORAL ALLOCATION OF GROUNDWATR TO AGRICULTURAL ACTIVITIES1

ABSTRACT: In the past, researchers have applied a variety of analytical techniques for maximizing the present value of a stock resource - simulation, calculus of variations, stochastic dynamic programming, and optimal control theory. This paper presents a more operational approach - linear programming. A simplified, broadly drawn example from Southwest agriculture is used for demonstrating the model's structure and output. The method is based on a set of state transformation operations that prevent the additivity assumption of linear programming from being violated.     

EFFECT OF URBANIZATION ON ALTERNATIVE FLOOD CONTROL STRATEGIES1

ABSTRACT: The effect of urbanization on alternative flood control strategies was investigated for a large developing watershed in Texas. Urban and rural areas were modeled separately using a geographically-referenced data base and the U.S. Corps of Engineers HEC-1 and HEC-2 programs, and results yielded a double-peaked hydrograph. Hydrograph input parameters were modified to predict the effects of a wide range of management alternatives including on-site storage, reservoirs, channelization, and development controls. Results indicated a combination of alternatives is required to protect existing and future developments.     

DEMAND SIGNALS FOR URBAN DRAINAGE AND FLOOD CONTROL PROJECTS1

ABSTRACT: Planning for urban drainage and flood control requires the use of a rational procedure for setting priorities and allocating funds. An innovative procedure developed by the Los Angeles County Flood Control District is described. It consists of the use of bond election voting results as a surrogate for demand signals representing the public preference for project approval. A regression equation has been developed to relate project characteristics to the likelihood of approval. The procedure is implemented through a “funnel-screen” review procedure which results in a list of reviewed and approved projects.