Linear Multiobjective Programming

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Numerical results showing viability of the algorithm proposed are presented. Abo-Sinna , A bilevel nonlinear multiobjective decision making under fuzziness, Journal of Operational Research Society of India , 38 , Google Scholar. Mansouri , An exact penalty on bilevel programs with linear vector optimization lower level, European Journal of Operational Research , , Benson , Optimization over the efficient set, Journal of Mathematical Analysis and Applications , 98 , Morgan , Semivectorial bilevel optimization problem: Penalty approach, Journal of Optimization Theory and Applications , , Gale , Linear bilevel programs with multiple objectives at the upper level, Journal of Computational and Applied Mathematics , , Savard , An overview of bilevel optimization, Annals of Operations Research , , Sinha , Solving bilevel multi-objective optimization problems using evolutionary algorithms, Lecture Notes in Computer Science: Evolutionary Multi-criterion Optimization , , Dempe , Annotated bibliography on bilevel programming and mathematical programs with equilibrium constraints, Optimization , 52 , Dutta , Is bilevel programming a special case of a mathematical program with complementarity constraints?

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Penalty method-based equilibrium point approach for solving the linear bilevel multiobjective programming problem. A penalty function algorithm with objective parameters for nonlinear mathematical programming. Yanqin Bai , Chuanhao Guo. Doubly nonnegative relaxation method for solving multiple objective quadratic programming problems. A new method for strong-weak linear bilevel programming problem. A majorized penalty approach to inverse linear second order cone programming problems. Linear programming based optimality conditions and approximate solution of a deterministic infinite horizon discounted optimal control problem in discrete time.

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Local smooth representation of solution sets in parametric linear fractional programming problems. Xinmin Yang. On second order symmetric duality in nondifferentiable multiobjective programming. Mansoureh Alavi Hejazi , Soghra Nobakhtian. Optimality conditions for multiobjective fractional programming, via convexificators. Linear programming solutions of periodic optimization problems: approximation of the optimal control. Charles Fefferman. Interpolation by linear programming I. Paul B. Hermanns , Nguyen Van Thoai. Global optimization algorithm for solving bilevel programming problems with quadratic lower levels.

A DC programming approach for a class of bilevel programming problems and its application in Portfolio Selection. Linear programming bounds for unitary codes. Advances in Mathematics of Communications , , 4 3 : Higher-order symmetric duality in multiobjective programming with invexity. Xinmin Yang , Xiaoqi Yang. The negative sign preceding the deviational variable tries to minimize the deviational variables.


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By this way equal preference is given to both parts of the final objective function. The problem under consideration is how to allocate limited water resources among competing water demand sectors, while maintaining a balance in the system which maximizes the overall system benefits.

Therefore, to solve such a problem, a compromise has been established between the upper and lower level decision makers, and the objectives functions are: 5 6. In which, 7 8. Bi-level water allocation program 9. Subject to: 10 11 In addition to the constraints in Eqs 10 — 12 , an additional compromise constraint has been developed between the upper level and lower level programs to solve the multi-objective function of the bi-level decision makers, as follows: The minimum and normal seasonal demands are obtained by accumulating the minimum and normal monthly demands of each sector during the dry season Oct-Mar.

The value of ULB is the value of water per unit volume of water released from the reservoir for any sector and is constant for each sector [ 51 ]. NEB to various water users have been derived from the published studies [ 52 , 53 ]. The net economic benefits from water supplied to irrigation are calculated from the total benefits of crop production minus the total production cost and then divided by the total volume of water supplied to the crop. To determine the monthly economic benefits, the seasonal NEB is multiplied by the ratio of monthly water supplied to the total seasonal water supplied while the costs like fertilizer, labor, machinery etc.

The water supplied from the reservoir to the residences and the public and other offices in different municipalities is taken as the domestic water use sector and its benefits are estimated by using the inverse demand function [ 56 ]. This is calculated as the difference between the water use benefits, minus the installation and maintenance cost of the water conveyance system.

This difference is then divided by the volume of water supplied from the reservoir giving the NEB per unit volume of water use, as follows: The net benefits from the industrial sector are also estimated in the same way as those for the domestic sector, i. The ratio of the difference between the water use benefits, and the water conveyance system cost to the volume of water supplied from the reservoir, giving the NEB per unit volume of water used.

The results of the empirical studies [ 57 , 58 ] can also be used to calculate the net economic returns to water use in industrial and domestic sectors. In this study, the hydropower sector is considered as a non-competing sector as the water released from the reservoir passes through the turbine to generate the hydropower.

The NEB of the hydropower sector is computed as the ratio of the hydropower generation multiplied by the difference between the power selling price and the generation cost to the volume of water passing through the power plant, as follows: As there is no well-established method available to calculate the exact net economic benefits in the environment water sector, therefore the benefits from this sector are calculated as the costs of avoiding damages or replacing services of the infrastructure which can cause due to the salt water intrusion.

The water allocation for this sector is mainly to control the saltwater intrusion. The input data for reservoir operation model ROM includes the monthly inflows into the reservoir, the reservoir physical characteristic i. HEC-HMS model was run on the daily basis and the simulation has been carried out for a period of 20 years i. The observed inflow data are presented in S1 Table.

Fig 6 shows the river network of Swat River basin and the comparisons between the observed and model simulated flows. In automated calibration techniques the model calculates the optimized parameter values that could result the best fit between observed and simulated runoffs [ 59 ]. A good agreement between the observed and simulated flows was found at the Junction J-1, where the Gabral and Ushu rivers join the Swat River. As stated earlier, the available water in the dry season is less than the total water required to satisfy the demand of various water use sectors. The water demands of the various sectors have been pooled sector-wise and considered for water allocation.

To demonstrate the model applicability, the limited available water resources of , 93, 81, , and Mm 3 for the months October to March in the dry season have been allocated for the monthly minimum and normal demands DM min and DM nor. As a solution to the bi-level water allocation problem, the amount of water allocated WA m is between the minimum and normal demands. The normal demand of a particular sector is defined as the water demand which that sector needs to fulfill its water requirements, and the minimum demand is the amount of water which must be released to the sector to fulfill its minimum requirements.

However, the weights can be varied depending on the priority either given to the upper or lower level DMs. Further, the hydropower sector is not considered as a competing water demand sector because the water released to various sectors passes through the turbines for hydropower production. Therefore, hydropower is produced as a by-product. By assigning equal priorities to all sectors, Table 2 shows the detailed model results in allocating the limited water resources among competing water demand sectors in different months of a season.

The model performance is satisfactory in the monthly water allocation program as the water allocated WA m is between the minimum and normal demands of each sector in each month, as shown in Table 2. As the upper level DMs tend to distribute the water resources to the lower level water users based on the equity system, therefore, the upper level controls the water allocation program as it aims to provide sustainability to the water allocation program.

On the other hand, the lower level DMs try to maximize the individual sectoral benefits. When equal priorities are given to each water demand sector, the water shortage is distributed equally to all the sectors. When the stress is equally distributed among all the sectors in a water scarce season, each of the water demand sectors receive water less than its normal demand but can survive since its minimum demand is fulfilled. Table 2 shows that all the sectors receive more water than their minimum requirements but less than the normal demands.

The satisfaction rate of a particular water use sector is the ratio of the amount of water supplied to the normal demand by that sector in a selected month or season. The level of satisfaction is an indication of the percentage of the water demand fulfilled; the remaining percentage is the stress level. When equal preference is given to all sectors, none of the sectors is fully satisfied and the water shortage is equally distributed among the sectors. The accumulated net economic benefits of all the water demand sectors in different months of a season are shown in Fig 7.

In March, the maximum economic returns are the maximum amount of water allocated to various sectors in this month because the accumulated water demand of all the sectors is highest as compared to that in the other months for the selected season. In most of the practical cases, the priorities are different depending on the local conditions and social preferences, e. Therefore, different flexible scenarios have been developed by assigning priorities to the water demand sectors to evaluate the model applicability under different conditions.

As in some of the regions or areas, irrigation could be the prioritized sector, therefore the upper level decision makers try to maximize the water allocation for the irrigation sector. However, in other regions, domestic sector maybe the priority in order to maximize the economic returns.

Therefore, the priority is different for different area depending on the requirements of that particular area. In this study, four scenarios have been developed by assigning priorities to different water demand sectors, namely: irrigation, industry, domestic and environment. The developed scenarios give a wide spectrum of the situation to the local decision makers and allow them to further develop and analyze other scenarios which suit the situation and improve the water management in the water scarce regions.

The model has been applied to allocate water on a seasonal basis. The input parameters and results of the developed scenarios are compared and shown in Table 3 , which are discussed in subsequent sections. In this scenario, priority is given to the irrigation sector in order to maximize the water allocation to this sector.

In this case, the upper level DMs allocate the maximum amount of water to this sector while also making sure that the minimum water requirements of the other sectors are fulfilled. The model first fulfills the minimum requirements of all the sectors. After that, water is allocated according to the given priority, such as the irrigation sector. Then, the remaining water is allocated to the other sectors.

Fig 8 shows that in Scenario-I, the economic benefits produced by the irrigation sector is the highest, as compared to all the other developed scenarios. Furthermore, the economic benefits in this scenario are less than the benefits produced when the equal priority is given to all the sectors Table 2 as the economic value of irrigation water is less than those of the industrial and domestic sectors. In Scenario-II, the priority is given to the industrial sector for the optimal allocation of water resources based on bi-level programming.

As the upper level decision makers control the water allocation program by releasing the water to lower level water users. Therefore, in this scenario, the water is first allocated to the industrial sector. The performance of the BLMOLP model is also satisfactory in this scenario as all the sectors fulfill their minimum water requirements. In this scenario, the economic benefits to the industrial sector are the highest as compared to the other scenarios, as shown in Fig 8. This is because the economic value of the industrial water use is higher than those of the irrigation and environment sectors Table 2.

A third scenario has been developed by assigning priority to the domestic sector and its results are shown in Table 3. After that, the model allocates the water according to the assigned priority. As the prioritized sector in this scenario is the domestic sector, the model allocates the remaining water to this sector to meet its normal demand.

After meeting the normal demand of the domestic sector, if there is any water left, then the model allocates the water to the sector which has minimum water requirements and maximum economic benefits. In this way, the maximum economic returns are achieved with minimum water release. Hence, the domestic sector produced the maximum economic benefits in Scenario-III as compared to other scenarios, as shown in Fig 8. Table 2 shows the total economic benefits when equal priority has been assigned to all the sectors. In this scenario, priority has been given to the environment sector.

A certain amount of water is required to be released to the downstream river for salinity control and to protect the downstream river inhabitants. In some regions or areas, the environmental sector maybe the prioritized sector in order to maximize the economic returns by protecting the downstream river inhabitants and by salinity control, which could otherwise damage the river infrastructure installments.

In this study, the benefits from the environment sector are considered as the cost which could be saved by preventing the damages caused by salinity. Table 3 shows the results when priority is given to the environment sector. Fig 8 shows that the value of the benefits derived from the environment sector is the highest as compared to the other sectors. This is because the unit value of benefits from the environmental sector is the lowest among all the considered water demand sectors.

Bi-level programming issues are frequently found in the allocation of water resources among various water users [ 30 ]. The proposed model offers an insight into the economic, water supply and hydrologic interaction for water allocation to distinctive water users. In the present study, there are not only conflicts among the different water users but also between the water users and reservoir managers.

Solving linear bilevel multiobjective programming problem via exact penalty function approach

Consequently, BLMOLP was evolved to optimally allocate the water resources among competing water users for sustainable economic development. The developed BLMOLP model is applied to a single reservoir by aggregating the releases from the reservoir for water allocation. However, the model structure may be improved by way of integrating parallel stems and can be applied to complex water resource networks or the model can be run separately for each reservoir in the network.

The reservoir is fulfilling the irrigation water demands of the areas located right away downstream, however, the water shortages occur in the further downstream areas in the course of the dry seasons. The reservoir simulation and water allocation calculations were performed for a period of 20 years. However, the water allocation results are only shown for a dry season Oct-Mar with the AW of Mm 3 against the total normal demand of all water users of Mm 3.

In this study, water is allocated based on the equity based and priority-based systems. Equal priorities were assigned to each water user in the equity-based system, which is also the current water allocation practice. In the priority based system, different scenarios were evolved by assigning priorities to specific water users to illustrate the model applicability under different conditions so that a suitable scenario acceptable to stakeholders may be developed, analyzed and implemented objectively to the water situation in the basin [ 16 ].

When the equal priorities were assigned to each sector, BLMOLP maximizes the equity in the water allocation system and the maximization of economic returns of individual sectors maybe compromised. Therefore, in a water allocation system based on equal priorities optimizes the upper level decision making process but it might compromise the decisions by the lower level decision makers. In priority based water allocation system, when a sector given priority produces highest economic benefits and satisfaction rate among all the scenarios because the model first allocates the water to the prioritized sector then to the remaining sectors based on their demands and net economic returns and the results were found consistent with earlier studies [ 30 , 60 ] under these conditions.

Multi-objective linear programming

When a particular water user gains priority, it attains maximum economic returns and satisfaction level among all the other selected water users and these results were found consistent with the previous studies using SICCON technique in water allocation. Using the Gini coefficient, a power index and economic efficiency function, Hu et al. The results of the current study are found consistent with [ 61 ], in which a two-stage regional multi-water source allocation model was developed which was capable to optimize the water allocation framework for water resources managers and DMs along with the benefits for the individual water users.

Furthermore, similar results were found in a two-stage stochastic fractional programming TSFP method for planning of an agricultural water resources management system [ 62 ]. However, the techniques used in the present study are mathematically simple and easy to apply and, therefore, technology transfer is considered to be more effective. There are some limitations and challenges that need to be addressed in the water allocation model developed in this study through further investigations.

Multiobjective approach in plans for treatment of cancer by radiotherapy

NEB to water use for salinity control and navigation should be determined using other suitable approaches and methods instead of the replacement cost method adopted here. The study assumes a fixed net benefit irrespective of amount of water allocated to each of the use sectors.


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This is justified as the monthly variation in the water allocated to different sectors is very small [ 60 ]. Moreover, the weights given to different water users can affect the results significantly, however, this study considers the same weight to different water users for simplicity in allocating the water resources which is not realistic as different water users may have different priorities depending upon their importance in the society irrespective of the economic returns [ 16 , 48 ].

Therefore, in future studies varying weights maybe assigned to different water users depending upon the objective functions and their importance in the society when designing the water allocation system for a reservoir. In this study, a bi-level multi-objective model has been developed for the optimal water allocation under the heirachical structure. The model has been applied to the Swat River basin of Pakistan for an optimal allocation of AW among competing water use sectors, i.

Different techniques have been used to estimate the NEB to water use in irrigation, domestic, industrial, hydropower, and environmental salinity control sectors. The study analyzes the performance of developed water allocation model under two conditions, i. When equal priority is given to all the water demand sectors, the water allocated to each sector do not meet the normal demand of any sector because the AW is less than the total demand of all water users.

However, the minimum water requirements of all the sectors are accomplished. Furthermore, four scenarios have been developed by prioritizing the four water use sectors individually, i. The bi-level programming model developed in this study provides a higher motivation for water saving and alleviates the conflict between water demand and supply by introducing the concepts of satisfaction rate and economic benefits together.

Also, the BLMOLP model has the advantage in addressing the bi-level water allocation problem because of its fewer requirements for data collection and solution generation. Browse Subject Areas? Click through the PLOS taxonomy to find articles in your field. Abstract This paper presents a simple bi-level multi-objective linear program BLMOLP with a hierarchical structure consisting of reservoir managers and several water use sectors under a multi-objective framework for the optimal allocation of limited water resources.

Introduction The ever-increasing population growth and industrialization are putting constant pressure on water resources and it is more likely that the available water resources may not be able to meet the future water demands. Study area Swat River is one of the main rivers in Khyber Pakhtunkhwa province of Pakistan with a drainage area of 13, km 2 at Munda Dam site. Download: PPT.

Fig 2. Mean monthly river flows at Munda dam during — Materials and methods Conceptual framework The basic working principle and components of the model are shown in Fig 3. Fig 3. Optimization techniques The basic algorithm used for the optimal allocation of limited water resources to various sectors is based on the deterministic linear programming. Weighting technique WT. By grouping the individual objectives into one single objective function, the upper level and lower level decision making problems becomes one single decision-making problem and is given by the following equation: 1 Where Z is the optimal allocation values, G is the minimization or maximization function, n is the number of objectives, and z is the individual objective function.

These scores are converted to weights by the following relationship; 2 Where, e mj is the converted scores of the m th objective given by the j th expert; n d is the number of judges in the group. Each deviational variable forms the compromise-constraint in the standard form, as follows: 3 Subjected to 4 where f u x and f u x are the upper level and the lower level objective functions whose individual maxima are , respectively, w u , w l are the weights of the upper level and lower level DMs, respectively. Fig 4. Graphical illustration of compromise constraint approach. Objective functions and calculation procedure The problem under consideration is how to allocate limited water resources among competing water demand sectors, while maintaining a balance in the system which maximizes the overall system benefits.

Net economic benefits to different sectors The minimum seasonal demand DS min and the normal demand DS nor of different water use sectors along with the upper level benefits ULB and the lower level benefits LLB are given in Table 1. Table 1. Seasonal water demands and NEB of various water-use sectors. Fig 6. Stream network and direction of flow in Swat River basin. Results and discussions Equity based water allocation The BLMOLP model has been applied to the Swat River basin in Khyber Pakhtunkhwa province of Pakistan, for optimal allocation of limited water resources to four competing water use sectors: i.

Table 2. Monthly water allocation results with equal priorities assigned to each sector. Fig 7. Total benefits by all water users in different months based on equity. Priority based water allocation In most of the practical cases, the priorities are different depending on the local conditions and social preferences, e. Scenario-I: Irrigation priority. Fig 8. Comparison of economic benefits in different scenarios. Scenario-II: Industry priority. Scenario-III: Domestic priority. Scenario-IV: Environment priority. Discussions Bi-level programming issues are frequently found in the allocation of water resources among various water users [ 30 ].

Conclusions In this study, a bi-level multi-objective model has been developed for the optimal water allocation under the heirachical structure. Supporting information. S1 Table. Daily inflows data. References 1. Al Radif A. Integrated water resources management IWRM : an approach to face the challenges of the next century and to avert future crises. View Article Google Scholar 2. Iranian Journal of Agricultural Economics; ;8: 47— View Article Google Scholar 3. Priya N, Geetha G. J Comput Theor Nanosci. View Article Google Scholar 4. Optimal irrigation water allocation using a genetic algorithm under various weather conditions.

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Optimal water allocation model based on satisfaction and economic benefits. Int J Water. A model for optimal allocation of water to competing demands. The future of water resources systems analysis: Toward a scientific framework for sustainable water management. Water Resour Res. Received View Article Google Scholar Topics in Safety, Risk, Reliability and Quality. Water resource management using multi-objective optimization and rainfall forecast. Convergence Information Technology, International Conference on. An optimal water allocation model based on water resources security assessment and its application in Zhangjiakou Region, northern China.

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