Linear Programming Techniques Research Articles

Signal Processing with Machine Learning for Context Awareness in 5G Communication Technology

To meet users’ expectations for speed and reliability, 5th Generation (5G) networks and other forms of mobile communication of the future will need to be highly efficient, flexible, and nimble. Because of the expected density and complexity of 5G networks, sophisticated network control across all layers is essential. In this context, self-organizing network (SON) is among the essential solutions for managing the next generation of mobile communication networks. Self-optimization, self-configuration, and self-healing (SH) are typical SON functions. This research creates a framework for analyzing SH by exploring the impact of recovery measures taken in precarious stages of health. For this reason, our suggested architecture takes into account both detection and compensating operations. The system is broken down into some faulty states and the “fuzzy c-means” (FCM) approach is used to conduct the classifying. In the compensation process, the network is characterized as the Markov decision model (MDM), and the linear programming (LP) technique is implemented to find the most effective strategy for reaching a goal. Numerical findings acquired from a variety of situations with varying objectives show that the suggested method with optimized operations in the compensation stage exceeds the approach with randomly chosen actions.

Formulation, optimization of a poultry feed and analysis of spectrometry, biochemical composition and energy facts

Poultry feed plays a key role in the success and economic profitability of poultry production. When formulating an effective feed, its cost and nutritional quality to cover the essentials of poultry must be taken into account. For this reason we tried to make an optimal poultry feed in quality and price, so we collected samples for this study The collection was organized to cover as much variability as possible related to seasons, suppliers, origins. Analysis must be done to discover the biochemical compositions and nutritional energy of the primary trades that lead to a formula to produce poultry feed, this formula is realized by linear programming technique to manufacture feed for three categories of poultry (starter (0–4 weeks), grower (4–18 weeks) and layer chickens (18 weeks to culling)). we analyze the samples by the near infrared spectrometry (NIRS), and to confirm the spectrometric results, we do deferent chemical analysis. Finally we obtain a database that guides us to interpret the optimal percentages for the manufacture of poultry feed.

An overview of AC and DC microgrid energy management systems

<abstract> <p>In 2022, the global electricity consumption was 4,027 billion kWh, steadily increasing over the previous fifty years. Microgrids are required to integrate distributed energy sources (DES) into the utility power grid. They support renewable and nonrenewable distributed generation technologies and provide alternating current (AC) and direct current (DC) power through separate power connections. This paper presents a unified energy management system (EMS) paradigm with protection and control mechanisms, reactive power compensation, and frequency regulation for AC/DC microgrids. Microgrids link local loads to geographically dispersed power sources, allowing them to operate with or without the utility grid. Between 2021 and 2028, the expansion of the world's leading manufacturers will be driven by their commitment to technological advancements, infrastructure improvements, and a stable and secure global power supply. This article discusses iterative, linear, mixed integer linear, stochastic, and predictive microgrid EMS programming techniques. Iterative algorithms minimize the footprints of standalone systems, whereas linear programming optimizes energy management in freestanding hybrid systems with photovoltaic (PV). Mixed-integers linear programming (MILP) is useful for energy management modeling. Management of microgrid energy employs stochastic and robust optimization. Control and predictive modeling (MPC) generates energy management plans for microgrids. Future microgrids may use several AC/DC voltage standards to reduce power conversion stages and improve efficiency. Research into EMS interaction may be intriguing.</p> </abstract>

Design Methodology for Wireless Backhaul/Fronthaul Using Free Space Optics and Fibers

The recent increase in data rates for Free space optics (FSO) transmission technology, means they could be used for designing the backhaul/fronthaul for 5G and beyond cellular networks. The flexibility and cost-effectiveness provided by FSO are the primary reasons for the mobile operators to investigate the potential of the technology as a mobile backhaul/fronthaul. Unfortunately, the reliability of FSO links is weather dependent, especially if the link covers considerable distance. Optical fibers, on the other hand, are expensive but more reliable. Hence, optimally designing a hybrid network consisting of both fiber and FSO connections can bring in the cost-effectiveness of FSO as well as the robustness of fibers. In such a design, the more important links are connected using fibers while the links with higher tolerance towards failure are designed using FSO. Therefore, in this paper, we propose a hybrid FSO/fiber backhaul/fronthauling methodology for connecting wireless base stations (BSs) to the network core. We first formulate a mixed integer non-linear program (MINLP) for determining the number of splitter/FSO distribution points required in the network that optimally provide connectivity to the BSs. The MINLP is designed to identify the locations of the splitters/FSO distribution points as well. Thereafter, we solve the MINLP with the help of particle swarm optimisation (PSO) and mixed integer linear programming (MILP) techniques. We also derive a heuristic for solving the MILP. Finally, we propose another method for determining the number of splitters required in a relatively shorter time: K-means cluster based method. The results verify that the hybrid network is cost-effective while conforming to the data rate and reliability requirements of the links. The proposal allows evaluation of a seamless design solution of the hybrid network with practical time-complexity.

Formulation and nutritional properties of qualea-bird-meat-based complementary foods for children (6–23 months) in Tanzania using a linear programming technique

BackgroundLong-standing mitigation strategies have been used to address the issue of malnutrition in children, especially undernutrition in developing countries including Tanzania, however, the problem still exists. Development of instant complementary formulae based on readily available local foods using robust mathematical models, to fight malnutrition seems to be one of the best options. PurposeThe purpose of this study was to develop and evaluate the nutritional content of instant supplemental food formulations based on qualea bird's meat for infants between the ages of 6 and 23 months using a linear programming model. Basic procedureFour complementary meal formulations based on bird meat were developed using a linear programming model: control, only bird meat (CtrQM); green banana mixed with bird meat (BanQ); Irish potato mixed with bird meat (PotQ); and rice mixed with the meat (RicQ). The formulations were cooked separately at 98 °C for 45–55 min and extruded to create instant dry products and analyzed using standard methods for proximate composition, minerals, and vitamin A. Main findingThe proximate composition parameters, minerals and vitamin A contents (in DM) were substantially enhanced and differed significantly (p < 0.05) between the formulations. PotQ had a higher moisture content (8.9 g/100 g), protein (16.7 g/100 g), crude fat (1.8 g/100 g), ash (5.4 g/100 g) and crude fibre (2.9 g/100 g) than lower values in other formulations. Furthermore, calcium was the most abundant mineral (832.5 mg/kg) whereas zinc was the least abundant mineral (38.5 mg/100 g). The variation in mineral contents among formulations was also significant (p < 0.05). Additionally, RicQ had significantly (p < 0.05) highest retinol (26.5 mg/kg) and beta-carotene (4.9 μg/g) while the lowest values were observed in the control sample with values of 3.3 mg/kg and 1.6 μg/g respectively. ConclusionIn a view of the findings, a linear programming model optimized and enhanced the proximate composition, mineral, and vitamin A qualities of the qualea birds' meat-based instant complementary formulation that seems to meet the RDAs for infants aged 6 to 23 months. Therefore, the application of the model in product formulation to combat malnutrition in the country is highly recommended.

Sensory properties, preference mapping and shelf life stability of linear programmed quelea meat based instant complementary food for children aged 6–23 months in Tanzania

Linear programming has been reported to be a good technique that can be used to develop and optimize complementary foods to curb children's malnutrition. Despite some studies using it to develop and assess the nutritional quality of linear programmed instant complementary foods based on high proteinous red-billed quelea bird meat, information on the sensory and shelf life stability of these foods is limited in the country. The objective of this study was to develop and assess the sensory properties, preference mapping and shelf life of the developed and optimized quale birds meat-based complementary flour and porridge. Four formulations; control containing birds meat only, banana mixed with meat (Banana_meat), potatoes mixed with meat (potatoes_meat) and rice mixed with meat (rice_meat) were developed by mixing with other ingredients and extruded separately using a barrel extrude to get flour. The extruded flours were assessed for shelf life and also used to develop porridge samples for sensory properties, consumer acceptance and preference mapping analyses using standard methods. There were significant (p < 0.05) differences in mean intensity scores between the formulation with potatoes mixed with meat samples having higher colour (7.5 ± 1.5), aroma (7.3 ± 1.6), viscosity (5.3 ± 3.0) and smoothness (7.65.3 ± 2.5) values than other samples with respective lower values of 4.4 ± 2.1–6.5 ± 1.5.2.6 ± 0.6–6.4 ± 2.4), 4.8 ± 0.2–5.2 ± 1.8 and 2.7 ± 0.6–6.3 ± 2.3. Rice mixed with meat sample had significantly highest whiteness (4.5 ± 1.5) and sweetness (7.5 ± 1.9) intensities scores while the control meat only samples had significantly highest oiliness intensity scores (5.0 ± 1.4) compared to other samples. Moreover, the potatoes mixed with meat sample was the most acceptable formulation (7.7 ± 0.3) compared to other samples with aroma, smoothness sweetness, colour and viscosity attributes being the main drivers for consumer liking of complementary porridge samples. The shelf life study revealed that the overall acceptability scores of complementary flour samples decreased progressively with storage days with the longest predicted shelf life of 140 days observed in rice mixed with meat sample followed by potatoes mixed with meat with 48 days and the shortest shelf life of 23.7 days in control samples. This suggests that quelea bird meat could be combined with cereal, tubers, and other minor ingredients like vegetables, optimized using the linear programming technique and extruded into complementary instant flour and porridge for children between the ages of 6 and 23 months with long shelf lives, appealing sensory properties, and consumer acceptance, respectively. Hence, it is advised to use the method and consume the developed formulations. • Linear programming resulted in complementary foods with appealing sensory properties and consumer acceptability. • The mean intensity score of sensory attributes differed significantly (p < 0.05) between the linear programmed complementary porridge samples. • Control (meat only) and potatoes_meat samples were the least and most acceptable formulations respectively. • Aroma, sweetness, smoothness, and colour were the main attributes driving consumers' acceptance of the porridge samples. • Rice_ meat and control samples had the longest and shortest shelf lives of 140 and 23.7 days respectively.

PRIVATIZING INDIAN AIRPORTS - A SCRUTINIZED EVALUATION THROUGH VARIED CONTEMPLATIONS

This paper elucidates the changing ownership of Public owned airports to Privatisation. The analysis is envisioned towards the passengers’ interest in availing air transport which are likely to become more efficient. The improvisation of regional connectivity would yield more people to opt for air transport. A series of efforts were taken before making the questionnaires that were placed in the survey including the psychological aspects when deciding about the choice of airports which are getting privatized. The collected inputs from the passengers vary from seldom travelers to frequent travelers across the country are analyzed using the linear programming technique such as the DEA[1] which depicts the effective views of the travelers to go for Privatisation. The questionnaire was carefully framed such that all the aspects covering ergonomics level, including the ease of navigation, the reach of supporting staff. The airports would get diversified with their services without the intervention of Government. In India, a total of 137 airports which include 23 International airports (3 Civil Enclaves), 10 Customs Airports (4 Civil Enclaves) having custom and immigration facilities for limited international operations and 104 Domestic airports (23 Civil Enclaves) are administered by the Airport Authority of India (AAI).[10] The USA has 14,549 Private airports (74%) within the total 19,636 airports. Many countries like the USA, Russia, China, U. K, France have a larger number of Privately-owned airports helping their countries in getting economically strengthened. IATA forecasts a growth of 6.8% per annum for India. The usage of No frill airports would reduce the cost of capital investment in making an airport to start functioning and help the nation to grow in the economy. In addition, Six AAI airports are operated, managed, and developed under Public-Private Partnership (PPP) namely, Guwahati, Jaipur, Ahmedabad, Mangalore, Thiruvananthapuram and Lucknow. At Mumbai, Delhi, Chennai, and Ahmedabad, Performance Based Navigation (PBN) arrival and departure procedures have been implemented and for Hyderabad airport, it is in progress and Space-based, Final Operational Phase (FOP) of (GAGAN), GPS Aided Geo Augmented Navigation[10] is under implementation which is not airport-based and will be available for use at all of the country's airports. Enhancing the prediction of outcomes through feature selection, the Data envelopment analysis (DEA) method is used for the analytics questionnaire. This is a function whose form is based on the efficiency of the scrutinized evaluations. It's an approach for measuring the performance efficiency of global public airports through varied contemplations of results of the survey. This technique aims to measure how efficiently a DEA uses the resources available to generate a group of outputs along with feature section models that are then analyzed after that to evaluate the contributing factors for airport operation and privatization.

A new integrated circular economy index and a combined method for optimization of wood production chain considering carbon neutrality

The wood industry is potentially advantageous to applying the concepts of circular economy for sustainable development and can contribute to the commitment of carbon neutrality. This study developed an integrated circular economy index based on five different quantitative indicators for assessment of the wood production chain: heat recovery rate, CO2 sequestration rate, fossil fuel substitution rate, renewable electricity usage rate, and revenue increase from the by-products. A combination of best-worst method (BWM) and linear goal programming (LGP) techniques was investigated to develop an optimal circular economy model of wood processing chain for reduction in CO2 emission. The integrated circular economy index and the combined method were tested in a case-study of a rubberwood processing chain in Vietnam. The proposed model suggests that the woodchips and biomass from the harvesting and processing of rubberwood could be collected and treated using microwave thermolysis techniques; the enzyme hydrolysis technique is appropriate for bioethanol and biomethane recovery from the sawdust; and the hot air technique is preferable in the drying process. The proposed model could result in a significant reduction of the total net carbon emission from +552,750 tons CO2eq to −1,145,940 tons CO2eq per year. This could support the achievement of Vietnam's zero CO2 emission goal and hopefully contribute to the country's commitment to carbon emission neutrality by the year 2050.

Efficiency evaluation of sewage treatment plants in Delhi, India, using tolerance-based data envelope analysis.

Correct and effective performance evaluation of wastewater treatment plants is a tough task because of the complex biological, physico-chemical, and biochemical processes and associated variables affecting their performance. Conventionally, the efficiency of sewage treatment plants (STPs) are obtained using some index relating pollutant removal efficiency with energy used or costs. These indicators consider only one variable at a time. This leads to incorrect assessment of efficiency, which in turn could adversely affect decision-making of the regulatory authorities. The data envelope analysis (DEA) method utilizes a Linear programming technique which can handle multiple input/output variables without requiring the cost function. This makes it an appropriate tool for assessing the relative efficiency of treatment plants. The present study assess the efficiency of 30 STPs in Delhi, India, using the tolerance-based DEA model utilizing the variable return of scale (VRS). The uncertainty was incorporated into the model using the tolerance measure. The model is solved using the "Add on" option in spreadsheet toolbox of excel solver. Results reveal that out of the 30 plants considered for the study, 6 are performing well (20%). Further, it was observed that a slight change in the input data leads to instability of the efficiency results. Lastly, the ranking is used to determine the treatment plant with best efficiency under all scenarios for the larger period of the year. Such studies will help in chalking out the best management practices that could be adopted by other regulatory authorities.

Comparison of Grid Reactive Voltage Regulation with Reconfiguration Network for Electric Vehicle Penetration

Renewable energy sources and EV growth brings new challenges for grid stabilization. Smart grid techniques are required to reconfigure and compensate for load fluctuation and stabilize power losses and voltage fluctuation. Numerical tools are available to equip the smart grid to deal with such challenges. Distribution Feeder reconfiguration and reactive voltage injection to the disturbed grid are some of the techniques employed for the purpose. However, either reconfiguration or injection alone is used commonly for this purpose. In this study, both techniques are applied to EV penetration as load and compared. A balanced IEEE 33 Radial network is used in this study and selected branches with high power losses are targeted for the reactive voltage injection and Minimum Spanning tree techniques (MST). EV charging loads are usually modelled with time base distribution which requires times base power flow analysis for reactive power injection. A comparison between coordinated, reconfiguration, and reactive voltage injection shows differences in power losses, voltage distortion, and cost saving. The analysis is carried out with an integer linear programming technique for coordinated charging, a minimum spanning tree for network reconfiguration, and genetic optimization for reactive power injection. Besides, all power flow analyses are carried out with the Backward/Forward sweep method. The information would help lowering power losses, grid stabilization, and charging station infrastructure planning.

Modeling of Polyculture Fish Feed Using Mathematical Programming

This study incorporates the application of a linear programming technique for composition of fish feed mixture to minimize the cost without sacrificing the nutrients levels. Designing the best fish diet for fry, fingerling, and growth stages is the main goal of this study. Regarding nutrient composition and manufacturing costs, the study's results show that a variety of elements can be combined much more effectively than in commercially available feed. The combinations lead to significantly reduced costs while raising nutrient intake, which might be quite advantageous to neighboring fish farmers. . The suggested model was solved with Microsoft Excel to make it more user-friendly.

A fairness-concern-based LINMAP method for heterogeneous multi-criteria group decision making with hesitant fuzzy linguistic truth degrees

Heterogeneous multi-criteria group decision making (MCGDM) is a hot topic in the decision analysis field. This paper proposes a fairness-concern-based LINMAP (Linear Programming Technique for Multidimensional Analysis of Preference) method for heterogeneous MCGDM with hesitant fuzzy linguistic (HFL) truth degrees. Heterogeneous evaluation information includes crisp numbers, interval numbers, intuitionistic fuzzy values (IFVs), trapezoidal fuzzy numbers (TrFNs) and hesitant fuzzy sets (HFSs). This paper introduces the fairness concern to calculate the HFL consistency and the HFL inconsistency indices. Based on the framework of LINMAP, a bi-objective HFL programming model is built to derive the criteria weights, the positive ideal fairness vector (PIFV) and the negative ideal fairness vector (NIFV) for each decision maker (DM) simultaneously. Based on the TOPSIS (Technique for Order Preference by Similarity to an Ideal Solution), a multi-objective programming model is built to obtain DMs’ weights. The alternatives ranking is derived by comprehensive collective relative closeness degrees. Finally, a real example is applied to verify effectiveness and superiority of this heterogeneous MCGDM method. The proposed heterogeneous MCGDM method provides a very useful approach for MCGDM with heterogeneous information.

High-Dimensional Data Cubes

This paper introduces an approach to supporting high-dimensional data cubes at interactive query speeds and moderate storage cost. The approach is based on binary(-domain) data cubes that are judiciously partially materialized; the missing information can be quickly reconstructed using statistical or linear programming techniques. This enables new applications such as exploratory data analysis for feature engineering and other fields of data science. Moreover, it removes the need to compromise when building a data cube - all columns that we might ever wish to use can be included as dimensions. Our approach also speeds up certain dice, roll-up, and drill-down operations on data cubes with hierarchical dimensions compared to traditional data cubes.

Event-triggered controller design for positive T-S fuzzy systems with random time-delay

This paper is concerned with the design of event-triggered controller for positive Takagi-Sugeno (T-S) fuzzy systems with a random time-delay. The random time-delay is described as a Markov process. A controller switched at different event-triggered instant is proposed. By constructing a new event-triggered instant-dependent linear co-positive Lyapunov function, the design criteria of event-triggered controller is derived to ensure the positivity and stability of the closed-loop system. These criteria can be solved by linear programming (LP) technique. A positive lower bound on the inter-execution time is ensured, which means that there is Zeno-free phenomenon. Finally, the simulation has demonstrated the effectiveness and merit of the proposed results.

Energy and economic assessment of a hybrid solar/gas heating system using a combined statistical-based multi-objective optimization method

The hybrid solar/gas heating systems are considered as promising technologies to reduce energy consumption and ecological impact, for a more sustainable environment. However, the complexity related to the process of hybridization often calls for multi-objective optimization problems to deal with the constraining objectives. In this paper, a multi-objective optimization methodology combining genetic algorithm, design of experiments and dynamic simulations is proposed. The present work is carried out to maximize the primary energy saving ratio (PESR), solar fraction (SF) and to minimize the levelized cost of heat of a hybrid solar/gas heating system intended for an office building located in Algiers, Algeria. The performance analysis of three control strategy modes showed how important the control strategies can improve the SF and the PESR. The suggested control mode demonstrated a seasonal SF and PESR of 40.31% and 31.32%, respectively. The proposed multi-objective optimization approach gave a better comprehensive energy and economic performance by identifying the pareto fronts. The Linear Programming Technique for Multidimensional Analysis of Preference decision-making technique was employed to determine the best optimal solution. The optimized system exhibited SF of 72.65%, PESR of 42.07% and the Levelized Cost of Heat of 0.054 $/kWh as per the design parameters, collector area of 10 m2, tank volume of 1.7 m3 and a flow rate of 0.2 kg s−1. Overall, the results showed that the proposed methodology is time-efficient and can be applied to optimize hybrid solar/gas systems.

Event‐triggered positive L1 filter design for positive Markovian jump systems with quantization

AbstractThis paper is concerned with the design of positive filter for positive Markovian jump systems (PMJSs) with adaptive event‐triggered mechanism and quantization. Considering the mode jump of the system, a new adaptive event‐triggered condition is established, which not only can regulate the communication threshold according to the error of dynamic systems, but also can avoid the asynchronous phenomenon between the system mode and the filter mode. In order to further reduce the burden of communication, a new positive filter with the quantized measure system output is developed, and the design condition which ensures the filter error system to be stochastically stable with performance is presented which can be solved in terms of linear programming (LP) technique. It is illustrated that there is no Zeno phenomenon. The proposed results are applied to the power allocation in a mobile telecommunication system, and it is shown that compared with the existing results, the event‐triggered filter of this paper has some advantages.

Image thresholding segmentation based on weighted Parzen-window and linear programming techniques

Image segmentation by thresholding is an important and fundamental task in image processing and computer vision. In this paper, a new bi-level thresholding approach based on weighted Parzen-window and linear programming techniques is proposed to use in image thresholding segmentation. First, by proposing a weighted Parzen-window to describe the gray level distribution status, we obtain the boundaries for the foreground and background of the image. Then the image thresholding problem can be transformed into the problem of solving a linear programming problem for computing the coefficient values of weighted Parzen-window. The results of testing on synthetic, NDT and a set of benchmark images indicate that the proposed method can achieve a higher segmentation accuracy and robustness in comparison to some classical thresholding methods, such as inter class variance method (OTSU), Kapur’s entropy-based method (KSW), and some state-of-art methods that consider spatial information, such as CHPSO, GLLV histogram method and GABOR histogram method.

Efficient Online City-Scale Patrolling by Exploiting Offline Model-Based Coordination Policy

With the increasing need of protecting public security, this paper studies the city-scale patrolling (CSP) problem, where hundreds of police officers are planned to patrol thousands of regions in a city. Given the stochastic nature with uncertain incident occurrence and travel time, the online CSP, where the patrolling policy should be determined sequentially, is of special interest. The online CSP aims at the omnipresence patrolling, which denotes that there are always police officers nearby that can respond timely when an incident occurs. Existing exact combinatorial optimization approaches are time-consuming and cannot scale to online CSP scenarios. On the other hand, within the dynamic CSP environments, existing decentralized multiagent coordination-based approaches always converge to the suboptimal solution without any performance guarantee. To achieve the omnipresence patrolling in a real-time fashion, a novel two-stage CSP framework is proposed. In the first stage, an offline model-based patrolling policy (OFFLINECSP) is designed, where the historical data is used to build the model and the linear programming (LP) technique is proposed for the coordination policy. Guided by the benchmark OFFLINECSP policy, an efficient online patrolling (ONLINECSP) policy is proposed in the second stage, where the police prefers to serve the critical incidents and to patrol hot-spot regions. The theoretical result shows that the competitive ratio of ONLINECSP can be guaranteed. Finally, extensive experiments on the synthetic and real datasets are conducted to validate the proposed framework. The results demonstrate that compared with existing benchmarks, the proposed two-stage CSP framework can not only maximize the incident service rate but also scale well to CSP in a real-time fashion.

Technical and economic modeling and optimization of a Ford-Philips engine for power production

In this study, for the first time, the effects of various parameters including engine rotational speed, engine pressure, heater temperature, and piston stroke on technical (energy and exergy) and economic (cost savings and payback period) performance of a Ford-Philips 4-215 engine is investigated. Considering the different engine losses, a thermodynamic model of the engine is employed and validated with available experimental data. The results reveal that the increase in engine pressure and heater temperature, augments the capacity and efficiency of the engine, respectively, and increasing the engine rotational speed and piston stroke first shortens the payback period and thereafter makes it longer. Moreover, the engine parameters are designed from six different technical and economic standpoints, and the optimal parameters are selected using two well-known decision methods, namely, the linear programming technique for multidimensional analysis of preference (LINMAP) and the technique for order preference by similarity to ideal solution (TOPSIS). In one of the cases, from energy and economic viewpoints, engine parameters have been designed and the maximum electric power and minimum payback period have been gauged at 260.46 kW and 6.10 years, respectively. In another case, from the exergy perspective, the maximum exergy efficiency and minimum exergy destruction are 65.17 % and 6.81 kW, respectively. Finally, by performing a three-objective optimization (energy, exergy, and economic), the maximum output power and exergy efficiency are evaluated 258 kW and 44.5%, respectively at a 5-year payback period.

Fuzzy Number Linear Programming Technique for Design of Rectangular Canals

Fuzzy Number Linear Programming Technique for Design of Rectangular Canals