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.     

FSILP: fuzzy-stochastic-interval linear programming for supporting municipal solid waste management

Although many studies on municipal solid waste management (MSW management) were conducted under uncertain conditions of fuzzy, stochastic, and interval coexistence, the solution to the conventional linear programming problems of integrating fuzzy method with the other two was inefficient. In this study, a fuzzy-stochastic-interval linear programming (FSILP) method is developed by integrating Nguyen's method with conventional linear programming for supporting municipal solid waste management. The Nguyen's method was used to convert the fuzzy and fuzzy-stochastic linear programming problems into the conventional linear programs, by measuring the attainment values of fuzzy numbers and/or fuzzy random variables, as well as superiority and inferiority between triangular fuzzy numbers/triangular fuzzy-stochastic variables. The developed method can effectively tackle uncertainties described in terms of probability density functions, fuzzy membership functions, and discrete intervals. Moreover, the method can also improve upon the conventional interval fuzzy programming and two-stage stochastic programming approaches, with advantageous capabilities that are easily achieved with fewer constraints and significantly reduces consumption time. The developed model was applied to a case study of municipal solid waste management system in a city. The results indicated that reasonable solutions had been generated. The solution can help quantify the relationship between the change of system cost and the uncertainties, which could support further analysis of tradeoffs between the waste management cost and the system failure risk.     

An inexact two-stage mixed integer linear programming method for solid waste management in the City of Regina

In this study, an interval-parameter two-stage mixed integer linear programming (ITMILP) model is developed for supporting long-term planning of waste management activities in the City of Regina. In the ITMILP, both two-stage stochastic programming and interval linear programming are introduced into a general mixed integer linear programming framework. Uncertainties expressed as not only probability density functions but also discrete intervals can be reflected. The model can help tackle the dynamic, interactive and uncertain characteristics of the solid waste management system in the City, and can address issues concerning plans for cost-effective waste diversion and landfill prolongation. Three scenarios are considered based on different waste management policies. The results indicate that reasonable solutions have been generated. They are valuable for supporting the adjustment or justification of the existing waste flow allocation patterns, the long-term capacity planning of the City's waste management system, and the formulation of local policies and regulations regarding waste generation and management.     

POLITICAL RESOLUTION OF ENVIRONMENTAL CONFLICTS1

ABSTRACT Many water resources management problems ultimately require a political decision in order to formulate a course of action to follow. Metagame analysis is a nonquantitative technique that can model the decision-making process and is used in this article to arrive at a politically feasible solution to a typical water pollution problem. The output from the linear programming formulation of the problem is used as input to the “political” metagame analysis. This allows a solution to the pollution conflict that is not only physically and economically feasible according to the linear programming analysis, but politically feasible as well.     

OPTIMIZATION OF WATER RESOURCES FOR IRRIGATION IN EAST JORDAN1

The paper describes an approach towards optimal allocation of surface and ground water resources to three agricultural areas in the Jordan Valley under conditions of scarce water supply. The optimizing model allocates water from three main rivers, each with reservoir storage, and from two ground water sources to three irrigation regions. Productivity of irrigation water, expressed as the net present value of the regional agricultural output, but allowing for crop water deficits, is first maximized using nonlinear programming. The allocation process then adopts techniques of linear programming to determine the least cost alternative based on the unit cost of water from each resource at each destination, as it varies with time.     

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.     

Integrated model for renewable energy planning in Turkey

All economic sectors are associated with energy use; therefore, government organizations aim to supply sustainable energy for human needs and economic growth. In particular, increased environmental concerns of the public in Turkey have impacted policymaking for renewable energy (RE) management in Turkey. The primary objective for RE sources of the Turkish Ministry of Energy is to ensure that 30% of the share of electricity production is from RE resources in 2023. In this paper, the integrated multi-objective, multi-period linear programming model is presented to determine effective allocation of RE supply for seven different geographical regions in Turkey for the period of 2017 to 2024. The integrated model consists of two different stages. The first stage involves qualitative evaluations of RE sources for seven geographical regions. Analytical Hierarchy Process (AHP) is applied to determine criteria priorities and overall ratings of geographical regions across determined criteria for RE sources are computed. The second stage of the integrated model consists of a multi-objective, multi-period linear programming model. The proposed multi-objective linear programming model is coded in MPL (Mathematical Programming Language) and solved using the GUROBI 5.1.0 solver. The output of the integrated model presents the total supply amount of RE sources for geographical regions in planning period. The ε-constraints method is applied to compute the total supply amount of RE from geographical regions for the period of 2017 to 2024. In this study, a systematic decision-making model is generated to allocate renewable energy sources to the geographical regions. The presented model integrates qualitative evaluations and quantitative parameters of different geographical regions to determine the optimal supply amount of RE. The obtained results are consistent with the potential quantities of RE alternatives in geographical regions, regional specifications, and social requirements.     

An interval type-2 fuzzy stochastic approach for regional-scale electric power system under parameter uncertainty

In this study, an interval type-2 fuzzy stochastic linear programming method (IT2FSLP) is developed to support regional-scale electric power system (REM) planning. The IT2FSLP-REM model is based on an integration of interval type-2 fuzzy sets boundary programming and stochastic linear programming techniques enable it to have robust abilities to the deal with uncertainties expressed as type-2 fuzzy intervals and probabilistic distributions within a general optimization framework. Moreover, it can reflect dynamic decisions for energy supply and energy conversion processes, as well as provide capacity expansion options with multiple periods. The developed model is applied to a case of planning regional-scale energy and environmental systems to demonstrate its applicability. Based on a two-step solution algorithm, reasonable solutions have been obtained, which reflect tradeoffs among economic cost, environmental requirements, and energy-supply security. Thus, the lower and upper solutions of IT2FSLP-REM would then help energy authorities adjust or justify allocation patterns of regional energy resources and services.     

UNIT HYDROGRAPHS – A COMPARATIVE STUDY1

ABSTRACT. Unit hydrographs derived by using two methods, linear programming and least squares, are compared. Test data comprise rainfall and runoff information from four storms over the North Branch Potomac River near Cumberland, Maryland. The mathematical bases of these methods for unit-hydrograph derivation are explained. The linear programming method minimizes the sum of absolute deviations, and the least squares method minimizes the sum of the squares of deviations. Computer subroutines are readily available for application of these methods. The unit hydrographs derived with the two methods are practically the same for storms 2 and 3, but differ somewhat for storms 1 and 4. However, the reconstituted direct surface runoff hydrographs are similar to those observed with the exception of the hydrograph for storm 4 which had a relatively more non-uniform rainfall excess of a considerably larger duration.     

FARM INCOME AND GROUND WATER QUALITY IMPLICATIONS FROM REDUCING SURFACE WATER SEDIMENT DELWERIES1

ABSTRACT: The potential surface water and ground water quality tradeoff implications from the nonpoint source provisions of the 1987 Water Quality Act are investigated in this paper using a national linear programming model developed at Iowa State University and modified by the Economic Research Service and the Leaching Evaluation of Agricultural Chemicals (LEACH) Handbook developed by the U.S. Environmental Protection Agency. The linear programming model is used to maximize net farm revenue using optimal combinations of crop rotations and tillage practices for each region of the United States given natural resource constraints. The LEACH handbook is used to determine the relative potential for pesticides to leach below the root zone for different soil types, hydrologic conditions, pesticides used, and tillage practices. The results indicate that imposing a surface water quality erosion constraint aimed at reducing sediment concentrations results in a larger decrease in farm income than imposing a uniform 5 ton per acre per year erosion constraint. Both constraints could result in regional improvement in ground water quality in some regions of the country while decreasing ground water quality in other regions.     

AN OPTIMIZATION MODEL FOR EFFICIENT MANAGEMENT OF URBAN WATER RESOURCES1

ABSTRACT: A mathematical programming model is proposed to determine economically efficient urban water resource allocation and pricing policy by maximizing the sum of the consumer and producer surplus. The optimization of this nonlinear problem is accomplished by the use of linear programming algorithm. The feasibility of using recycled water for municipal purposes is examined in a planning context. The impact of higher water quality discharge standards on pricing and allocation of water is analyzed and the attractiveness of water reuse option is demonstrated.     

AQUIFER MANAGEMENT UNDER TRANSIENT AND STEADY-STATE CONDITIONS1

ABSTRACT: The equations of transient and steady-state flow in two-dimensional artesian aquifers are approximated using finite differences. The resulting linear difference equations, combined with other linear physical and management constraints and a linear objective function, comprise a linear programming (LP) formulation. Solutions of such LP models are used to determine optimal well distributions and pumping rates to meet given management objectives for a hypothetical transient problem and for a steady-state field problem.     

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 LINEAR ANALYSIS OF AN URBAN WATER SUPPLY SYSTEM1

The use of linear programming as a planning tool for determining the optimal long-range development of an urban water supply system was explored. A stochastic trace of water demand was synthesized and used as an input to the model. This permitted evaluating the feasibility of imposing demand restrictions as an effective cost reduction mechanism. The City of Lincoln, Nebraska, was used as the urban model. The fundamental problem was to allocate limited water supplies from several sources to an urban load center to minimize costs and comply with system constraints. The study period covered twenty years, and findings indicate the planning direction for stage development during this period. Sensitivity analyses were performed on cost coefficients and demands. Thirteen sources were included in the initial computations. Conclusions were that linear programming and generated demand traces are useful tools for both short- and long-term urban water supply planning. Lowering peak demands results in long-range development of fewer sources of supply and more economic and efficient use of the supplies developed.     

Inexact fuzzy-stochastic mixed-integer programming approach for long-term planning of waste management – Part A: Methodology

In this study, an inexact fuzzy chance-constrained two-stage mixed-integer linear programming (IFCTIP) approach is proposed for supporting long-term planning of waste-management systems under multiple uncertainties in the City of Regina, Canada. The method improves upon the existing inexact two-stage programming and mixed-integer linear programming techniques by incorporating uncertainties expressed as multiple uncertainties of intervals and dual probability distributions within a general optimization framework. The developed method can provide an effective linkage between the predefined environmental policies and the associated economic implications. Four special characteristics of the proposed method make it unique compared with other optimization techniques that deal with uncertainties. Firstly, it provides a linkage to predefined policies that have to be respected when a modeling effort is undertaken; secondly, it is useful for tackling uncertainties presented as intervals, probabilities, fuzzy sets and their incorporation; thirdly, it facilitates dynamic analysis for decisions of facility-expansion planning and waste-flow allocation within a multi-facility, multi-period, multi-level, and multi-option context; fourthly, the penalties are exercised with recourse against any infeasibility, which permits in-depth analyses of various policy scenarios that are associated with different levels of economic consequences when the promised solid waste-generation rates are violated. In a companion paper, the developed method is applied to a real case for the long-term planning of waste management in the City of Regina, Canada.     

Waste management with recourse: An inexact dynamic programming model containing fuzzy boundary intervals in objectives and constraints

The existing inexact optimization methods based on interval-parameter linear programming can hardly address problems where coefficients in objective functions are subject to dual uncertainties. In this study, a superiority–inferiority-based inexact fuzzy two-stage mixed-integer linear programming (SI-IFTMILP) model was developed for supporting municipal solid waste management under uncertainty. The developed SI-IFTMILP approach is capable of tackling dual uncertainties presented as fuzzy boundary intervals (FuBIs) in not only constraints, but also objective functions. Uncertainties expressed as a combination of intervals and random variables could also be explicitly reflected. An algorithm with high computational efficiency was provided to solve SI-IFTMILP. SI-IFTMILP was then applied to a long-term waste management case to demonstrate its applicability. Useful interval solutions were obtained. SI-IFTMILP could help generate dynamic facility-expansion and waste-allocation plans, as well as provide corrective actions when anticipated waste management plans are violated. It could also greatly reduce system-violation risk and enhance system robustness through examining two sets of penalties resulting from variations in fuzziness and randomness. Moreover, four possible alternative models were formulated to solve the same problem; solutions from them were then compared with those from SI-IFTMILP. The results indicate that SI-IFTMILP could provide more reliable solutions than the alternatives.     

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.     

OPTIMAL GROUND WATER MANAGEMENT IN TWO-AQUIFER SYSTEMS1

ABSTRACT; This paper presents a numerical model for the prediction of optimal ground water withdrawal from a two-aquifer system by observing a set of constraints determined by the ecological conditions of the ground water basin. The aquifer system consists of an upper unconfined and a lower confined aquifer with a leaky stratum between them. It is assumed that water is withdrawn from the confined aquifer only, but the unconfined aquifer will also be affected due to the leakiness of the layer separating the upper and lower aquifers. Simulation and linear programming are employed for developing a computer model for the optimal management of such systems, with the objectives of determining withdrawal rates for predetermined ground water levels.     

Optimal Location and Management of Waste Disposal Facilities Affecting Ground Water Quality1

ABSTRACT: Numerical simulation of ground water solute transport is combined with linear programming to optimize waste disposal. A discretized form of the equation governing solute transport is included as a set of constraints in a linear program. Two problems are described. First, the management model is used to maximize ground water waste disposal. The model constrains disposal activities so that the quality of local ground water supplies is protected. Parametric programming is shown to be important in evaluating waste disposal tradeoffs at the various facilities. Changes in the velocity field induced by waste water injection cause a nonlinearity in the solute transport equation which is dealt with by employing an iterative procedure. The second problem is aimed at identifying all sites which are suitable for waste disposal in the subsurface. The management model is manipulated so that the optimal value of the dual variables are “unit source impact indicators.” This physical interpretation is valuable in identifying feasible disposal sites. The joint simulation and optimization approach permits the management of complex ground water systems where the aquifer is used simultaneously for waste disposal and water supply.