A Fuzzy Multi-Objective Stochastic Programming Model for Allocation of Lands in Agricultural Systems

In this chapter, a fuzzy stochastic multi-objective programming model is presented for planning proper allocation of agricultural lands in hybrid uncertain environment so that optimal production of several seasonal crops in a planning year can be achieved. In India, demands of various seasonal crops are gradually increasing due to rapid growth of population, whereas agricultural lands are gradually decreasing due to urbanization. Therefore, it is a huge challenge to the planners to balance this situation by proper planning for the utilization of agricultural lands and resources. From that viewpoint, the methodology is developed in this chapter. To make the model more realistic, the resource parameters incorporated with the problem are considered either in the form of fuzzy numbers (FNs) or random variables having fuzzy parameters. The two main objectives of this agricultural land allocation model are considered as maximizing the production of seasonal agricultural crops and minimizing the total expenditure by utilizing total cultivable lands in a planning period. These objectives are optimized based on the constraints: land utilization, machine-hours, man-days, fertilizer requirements, water supply, etc. As the parameters associated with the constraints are imprecise and uncertain in nature, the constraints are represented using either FN or fuzzy random variables (FRVs). The reasons behind the consideration of fuzzy constraints or fuzzy chance constraints (i.e., the reason for considering the parameters associated with the constraints as FNs or FRVs in the model) are clarified in detail. As a study region, the District Nadia, West Bengal, India is taken into account for allocation of land. To illustrate the potential use of the approach, the model solutions are compared with the existing land allocation of the district.

Fuzzy Multi-Objective Programming With Joint Probability Distribution

In this chapter, a fuzzy goal programming (FGP) model is employed for solving multi-objective linear programming (MOLP) problem under fuzzy stochastic uncertain environment in which the probabilistic constraints involves fuzzy random variables (FRVs) following joint probability distribution. In the preceding chapters, the authors explain about linear, fractional, quadratic programming models with multiple conflicting objectives under fuzzy stochastic environment. But the chance constraints in these chapters are considered independently. However, in practical situations, the decision makers (DMs) face various uncertainties where the chance constraints occur jointly. By considering the above fact, the authors presented a solution methodology for fuzzy stochastic MOLP (FSMOLP) with joint probabilistic constraint following some continuous probability distributions. Like the other chapters, chance constrained programming (CCP) methodology is adopted for handling probabilistic constraints. But the difference is that in the earlier chapters chance constraints are considered independently, whereas in this chapter all the chance constraints are taken jointly. Then the transformed problem involving possibilistic uncertainty is converted into a comparable deterministic problem by using the method of defuzzification of the fuzzy numbers (FNs). Objectives are now solved independently under the set of modified system constraints to obtain the best solution of each objective. Then the membership function for each objective is constructed, and finally, a fuzzy goal programming (FGP) model is developed for the achievement of the highest membership goals to the extent possible by minimizing group regrets in the decision-making context.

Introduction and Historical Background

This chapter describes the evolution of different multi-objective decision-making (MODM) models with their historical backgrounds. Starting from MODM models in deterministic environments along with various solution techniques, the chapter presents how different kinds of uncertainties may be associated with such decision-making models. Among several types of uncertainties, it has been found that probabilistic and possibilistic uncertainties are of special interests. A brief literature survey on different existing methods to solve those types of uncertainties, independently, is discussed and focuses on the need of considering simultaneous occurrence of those types of uncertainties in MODM contexts. Finally, a bibliographic survey on several approaches for MODM under hybrid fuzzy environments has been presented. Through this chapter the readers can be able to get some concepts about the historical development of MODM models in hybrid fuzzy environments and their importance in solving various real-life problems in the current complex decision-making arena.

Fuzzy Multi-Objective Stochastic Models for Municipal Solid Waste Management

With rising urbanization and change in lifestyle and food habits of human beings, the amount of municipal solid wastes (MSWs) are now being increasing day by day, and the composition of wastes are now also being changed. Therefore, it is now becoming essential to develop a consistent mathematical model for managing those wastes in a systematic manner. In this context, fuzzy chance constrained programming (FCCP) model becomes useful to handle wastes efficiently through the process of selecting sorting stations, treatment facilities, etc. through an efficient way so that the net system cost of sorting and transporting the wastes would be minimized, and the revenue generated from different sorting stations and different treatment facilities would be to maximized. From that view point, in this chapter, a fuzzy chance constrained programming (CCP) model is developed for MSW management. Most of the parameters involved with this model are imprecisely defined and probabilistically uncertain. So, the parameters of the objectives are considered as FNs, and the right side parameters of the probabilistic constraints involve normally distributed fuzzy random variables (FRVs). To resolve the cases arising due to the multiple occurrences of fuzzy goals, a fuzzy goal programming (FGP) has been adopted. To expound the potential use of the approach, a modified version of a case example, studied previously, is considered and solved. The achieved model solution is discussed elaborately to illustrate the proposed methodology for MSW management.

Fundamental Concepts

In this chapter, the authors discuss some basic concepts of probability theory and possibility theory that are useful when reading the subsequent chapters of this book. The multi-objective fuzzy stochastic programming models developed in this book are based on the concepts of advanced topics in fuzzy set theory and fuzzy random variables (FRVs). Therefore, for better understanding of these advanced areas, the authors at first presented some basic ideas of probability theory and probability density functions of different continuous probability distributions. Afterwards, the necessity of the introduction of the concept of fuzzy set theory, some important terms related to fuzzy set theory are discussed. Different defuzzification methodologies of fuzzy numbers (FNs) that are useful in solving the mathematical models in imprecisely defined decision-making environments are explored. The concept of using FRVs in decision-making contexts is defined. Finally, the development of different forms of fuzzy goal programming (FGP) techniques for solving multi-objective decision-making (MODM) problems is underlined.

Fuzzy Linear Multi-Objective Stochastic Programming Models

In this chapter, fuzzy goal programming (FGP) technique is presented to solve fuzzy multi-objective chance constrained programming (CCP) problems having parameters associated with the system constrains following different continuous probability distributions. Also, the parameters of the models are presented in the form of crisp numbers or fuzzy numbers (FNs) or fuzzy random variables (FRVs). In model formulation process, the imprecise probabilistic problem is converted into an equivalent fuzzy programming model by applying CCP methodology and the concept of cuts of FNs, successively. If the parameters of the objectives are in the form of FRVs then expectation model of the objectives are employed to remove the probabilistic nature from multiple objectives. Afterwards, considering the fuzzy nature of the parameters involved with the problem, the model is converted into an equivalent crisp model using two different approaches. The problem can either be decomposed on the basis of tolerance values of the parameters; alternatively, an equivalent deterministic model can be obtained by applying different defuzzification techniques of FNs. In the solution process, the individual optimal value of each objective is found in isolation to construct the fuzzy goals of the objectives. Then the fuzzy goals are transformed into membership goals on the basis of optimum values of each objective. Then priority-based FGP under different priority structures or weighted FGP is used for achievement of the highest membership degree to the extent possible to achieve the ideal point dependent solution in the decision-making context. Finally, several numerical examples considering different types of probability distributions and different forms of FNs are considered to illustrate the developed methodologies elaborately.

Fully Fuzzified Multi-Objective Stochastic Programming

This chapter develops a methodology for solving fully fuzzified multi-objective chance constrained programming (CCP) problems with fuzzy random variables (FRVs) as parameters. In the preceding chapters, it is assumed that the parameters of the multi-objective programming models are uncertain, and these uncertain parameters are expressed through fuzzy numbers (FNs) and FRVs. However, in practical situations, it is also observed that not only the parameters but also the variables of the multi-objective programming problems are uncertain. From that view point, the methodology for solving fully fuzzified multi-objective stochastic programming problems are presented in this chapter. At first the fuzzy probabilistic constraints are modified into fuzzy constraints. Using the defuzzification method of FNs, the different fuzzy parameters and fuzzy variables in the constraints are converted into crisp equivalent parameters and crisp variables. In this chapter, the parameters of the objectives are considered as either symmetric trapezoidal FNs or FRVs whose mean and variances are taken as symmetric trapezoidal FNs. If the parameters of the objectives are FRVs, then expectation model and variance model of the objective is used to find an equivalent form of the objectives whose parameters are only FNs. The ranking function of FNs is then applied to the objectives to convert them into crisp objectives. Then each objective is solved independently under the modified system constraints to construct the membership goals of each objective. Finally, weighted fuzzy goal programming (FGP) model is applied to achieve the most satisfactory solution for the overall benefit of the organization. Two illustrative numerical examples are given to demonstrate the efficiency of the proposed methodology and to compare the solution obtained by the developed methodology with the pre-defined techniques.

Methodology for Solving Multi-Objective Quadratic Programming Problems in a Fuzzy Stochastic Environment

This chapter presents two methodologies for solving quadratic programming problems with multiple objectives under fuzzy stochastic environments. The right side parameters of the chance constraints of both the models are chosen as fuzzy random variables (FRVs) following different probability distributions. Like the previous chapters, chance constrained programming (CCP) methodology is employed to the fuzzy chance constraints to develop fuzzy programming model. In the first model, cut of fuzzy sets and fuzzy partial order relations are incorporated to the fuzzy programming model to develop an equivalent deterministic model. For the second model, defuzzification method of fuzzy numbers (FNs), which are presented in Chapter 2, are taken into consideration to generate equivalent quadratic programming model in a crisp environment. As the objective functions are quadratic in nature, it is easy to understand that the membership functions obtained through methodological development process are also quadratic in nature. To linearize the quadratic membership functions, linearization techniques are employed in this chapter. Finally, for achieving the maximum degree of each of the membership goals of the objectives, a fuzzy goal programming (FGP) approach is developed for the linearized membership goals and solved by minimizing under-deviational variables and satisfying modified system constraints in fuzzy stochastic decision-making environments. To illustrate the acceptability of the developed methodology presented in this chapter, some numerical examples are included.

Development of Fuzzy Multi-Objective Stochastic Fractional Programming Models

In this chapter, two methodologies for solving multi-objective linear fractional stochastic programming problems containing fuzzy numbers (FNs) and fuzzy random variables (FRVs) associated with the system constraints are developed. In the model formulation process, the fuzzy probabilistic constraints are converted into equivalent fuzzy constraints by applying chance constrained programming (CCP) technique in a fuzzily defined probabilistic decision-making situation. Then two techniques, -cut and defuzzification methods, are used to convert the model into the corresponding deterministic model. In the method of using -cut for FNs, the tolerance level of FNs is considered, and the constraints are reduced to constraints with interval coefficients. Alternatively, in using defuzzification method, FNs are replaced by their defuzzified values. Consequently, the constraints are modified into constraints in deterministic form. In the next step, the constraints with interval coefficients are customized into its equivalent form by using the convex combination of each interval. If the parameters of the objectives are triangular FNs, then on the basis of their tolerance ranges each objective is decomposed into three objectives with crisp coefficients. Then each objective is solved independently to find their best and worst values and those values are used to construct membership function of each objective. Finally, the compromise solution of multi-objective linear fractional CCP problems is obtained by applying any of the approaches: priority-based fuzzy goal programming (FGP) method, Zimmermann's approach, -connective process, or minimum bounded sum operator technique. To demonstrate the efficiency of the above-described techniques, two illustrative examples, studied previously, are solved, and the solutions are compared with the existing methodology.

On Solving Fuzzy Multi-Objective Multi-Choice Stochastic Transportation Problems

This chapter presents solution procedures for solving unbalanced multi-objective multi-choice stochastic transportation problems in a hybrid fuzzy uncertain environment. In this chapter, various types of unbalanced multi-objective fuzzy stochastic transportation models are considered with the assumption that the parameters representing supplies of the products at the origins and demands of the products at the destinations, capacity of the conveyances, associated with the system constraints are either fuzzy numbers (FNs) or fuzzy random variables (FRVs) with some known continuous fuzzy probability distributions. The multi-choice cost parameters are considered as FNs. In this chapter, two objectives are considered: total transportation cost and total transportation time. As the transportation cost mainly depends on fuel prices and since fuel prices are highly fluctuating, the cost parameters are taken as multi-choice cost parameters with possibilistic uncertain nature. The time of transportation mainly depends on vehicle conditions, quality of roads, and road congestion. Due to these uncertain natures, the parameters representing time of transportation are also taken as fuzzy uncertain multi-choice parameters. In this transportation model, these objectives are minimized satisfying the constraints: product availability constraints, requirement of the product constraints, and capacity of the conveyance constraints. Numerical examples are provided for the sake of illustration of the methodology presented in this chapter, and also achieved solutions are compared with the solutions obtained by some existing methodologies to establish its effectiveness.