Stochastic Programs with Recourse

The study of making the best decision under risk management in a variety of areas of our lives is known as Stochastic Programming. We will go through two-stage Stochastic Linear Programming approaches for a variety of real-world choice issues, as well as how to solve them. We will achieve this by constructing stochastic linear programming models based on real-world situations like the well-known Farmer's situation and News Vendors problems. The influence of pricing, Stochastic Integer Linear Programming problem, second stage Stochastic Integer Linear Programming problem, first stage Stochastic Binary Linear Programming problem, risk aversion problem, and continuous function for random variables based on two-stage SLP with the aid of Farmer's problem will all be examined. We will address the Newsvendor’s problem with Deterministic Equivalent Stochastic Linear Programming, an extension of Deterministic Stochastic Linear Programming for risk aversion with a high number of decision variables and restrictions, utilizing the two-stage Stochastic Linear Programming approach once more. Hand calculation is a challenging way to acquire the solution to the problems. As a result, we will use the programming language AMPL to design computer solutions for tackling both farmer and newsvendor difficulties. We will also utilize MATLAB to create graphs for the farmer's problem's continuous function. Dhaka Univ. J. Sci. 72(1): 30-45, 2024 (January)

Marine fisheries play an important role in the economic development of Indonesia. Besides being the most affordable source of animal protein in the diet of most people in the country, this industrial sector could provide employment to thousands who live at coastal area. We consider the management of small scale traditional business at North Sumatera Province which processes fish into several local seafood products. The inherent uncertainty of data (for example, demand and fish availability), together with the sequential evolution of data over time leads the production planning problem to be a linear mixed-integer stochastic program. We use a scenario generation based approach for solving the model. The result shows the amount of each fish processed product and the number of workforce needed in each horizon planning. References Birge J. R., Louveaux F. V. Introduction to stochastic programming. New York: Springer; 1997. Mawengkang H., Saib Suwilo, Opim S. Sitompul. Revision Modeling of Two-Stage Stochastic Programming Problem. Journal of Industrial System 2006; 7 (4):6--10. Mawengkang H., Suherman. A Heuristic Method of Scenario Generation in Multi-Stage Decision Problem under Uncertainty. Journal of Industrial System 2007; 8(2) : 98--105. Mulvey J. M., Van derbei R., Zenios S. Robust optimization of large scale systems. Operations research 1995;43 (2): 264--281. Rico-Ramirez V. Two-stage stochastic linear programming: a tutorial. SIAG/OPT Views-and-News 2002; 13 (1): 8--14. Sen S., Higle J. L. Introductory tutorial on stochastic linear programming models. Interfaces 1999; 29 (2): 33--61. Van der Vlerk M. H., Haneveld W. K. Stochastic integer programming: General models and algorithms. Annals of Operations Research,85:39--57,1999.

I. The General Framework.- 1. Multiobjective programming under uncertainty : scope and goals of the book.- 2. Multiobjective programming : basic concepts and approaches.- 3. Stochastic programming : numerical solution techniques by semi-stochastic approximation methods.- 4. Fuzzy programming : a survey of recent developments.- II. The Stochastic Approach.- 1. Overview of different approaches for solving stochastic programming problems with multiple objective functions.- 2. "STRANGE" : an interactive method for multiobjective stochastic linear programming, and "STRANGE-MOMIX" : its extension to integer variables.- 3. Application of STRANGE to energy studies.- 4. Multiobjective stochastic linear programming with incomplete information : a general methodology.- 5. Computation of efficient solutions of stochastic optimization problems with applications to regression and scenario analysis.- III. The Fuzzy Approach.- 1. Interactive decision-making for multiobjective programming problems with fuzzy parameters.- 2. A possibilistic approach for multiobjective programming problems. Efficiency of solutions.- 3. "FLIP" : an interactive method for multiobjective linear programming with fuzzy coefficients.- 4. Application of "FLIP" method to farm structure optimization under uncertainty.- 5. "FULPAL" : an interactive method for solving (multiobjective) fuzzy linear programming problems.- 6. Multiple objective linear programming problems in the presence of fuzzy coefficients.- 7. Inequality constraints between fuzzy numbers and their use in mathematical programming.- 8. Using fuzzy logic with linguistic quantifiers in multiobjective decision making and optimization: A step towards more human-consistent models.- IV. Stochastic Versus Fuzzy Approaches and Related Issues.- 1. Stochastic versus possibilistic multiobjective programming.- 2. A comparison study of "STRANGE" and "FLIP".- 3. Multiobjective mathematical programming with inexact data.

Solving stochastic programming problems using new approach to Differential Evolution algorithm

The concept of ranking method is an efficient approach to rank fuzzy numbers. In this paper, we have studied stochastic fuzzy multiobjective linear fractional programming problem (SFMOLFPP) where SFMOLFPP is transformed to its equivalent deterministic-crisp multiobjective linear programming problem (MOLPP). To study SFMOLFPP, a SFMOLFPP is presented in which the fuzzy coefficients and scalars in the linear fractional objectives and the fuzzy coefficients are characterised by triangular or trapezoidal fuzzy numbers. The left hand side of the stochastic fuzzy constraints are characterised by triangular or trapezoidal fuzzy numbers, while the right hand sides are assumed to be independent random variable with known distribution function. We have modify Iskander's approach [16] to transform the suggested problem to its equivalence deterministic-crisp MOLPP. We have also used ranking function in SFMOLFPP to find the pareto optimal solution of the reduced multiobjective linear fractional programming problem (MOLFPP). One numerical example is presented to demonstrate two methodologies.

Contributors

Numerous multiobjective linear programming methods have been proposed in the last two decades for contexts where the parameters are deterministic. In many real situations, however, parameters of a stochastic nature arise and the analyst is, as a result, confronted with a stochastic multiobjective linear programming problem. Recently, some methods have been developed to deal with this kind of problems; in most of them, the decision maker is supposed to be placed in a risky situation, i.e. a situation where he can associate probability distributions to the stochastic parameters. In many cases, we believe that it would be more realistic to suppose that the decision maker is in a situation of partial uncertainty, i.e. a situation where he possesses only an incomplete information about the stochastic parameters; for example, he could be able to precise only the bounds of variation of the parameters and eventually, their central values. For such situations, we propose a general multiobjective stochastic linear programming methodology which includes many modes of transformation of the stochastic objectives and constraints in order to obtain an equivalent multiobjective deterministic linear programming problem. Finally, this deterministic equivalent program is solved by an interactive method derived from STEM. Our methodology will be illustrated through a didactical example.

ABSTRACTThe computational complexity of linear and nonlinear programming problems depends on the number of objective functions and constraints involved and solving a large problem often becomes a difficult task. Redundancy detection and elimination provides a suitable tool for reducing this complexity and simplifying a linear or nonlinear programming problem while maintaining the essential properties of the original system. Although a large number of redundancy detection methods have been proposed to simplify linear and nonlinear stochastic programming problems, very little research has been developed for fuzzy stochastic (FS) fractional programming problems. We propose an algorithm that allows to simultaneously detect both redundant objective function(s) and redundant constraint(s) in FS multi-objective linear fractional programming problems. More precisely, our algorithm reduces the number of linear fuzzy fractional objective functions by transforming them in probabilistic–possibilistic constraints characterized by predetermined confidence levels. We present two numerical examples to demonstrate the applicability of the proposed algorithm and exhibit its efficacy.

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Denardo and B. L. July 2006 | SIAM Journal on Applied Mathematics, Vol. No. 3AbstractPDF of Stationary Optimal Policies for Some Markov July 2006 | SIAM Review, Vol. 9, No. 3AbstractPDF Programming by Linear August 2006 | SIAM Journal on Applied Mathematics, Vol. 14, No. 9, August July 2006 for and Applied & for and Applied Mathematics

A general concept for solving linear multicriteria programming problems with crisp, fuzzy or stochastic values

Most of the real-life decision-making problems have more than one conflicting and incommensurable objective functions. In this paper, we present a multiobjective two-stage stochastic linear programming problem considering some parameters of the linear constraints as interval type discrete random variables with known probability distribution. Randomness of the discrete intervals are considered for the model parameters. Further, the concepts of best optimum and worst optimum solution are analyzed in two-stage stochastic programming. To solve the stated problem, first we remove the randomness of the problem and formulate an equivalent deterministic linear programming model with multiobjective interval coefficients. Then the deterministic multiobjective model is solved using weighting method, where we apply the solution procedure of interval linear programming technique. We obtain the upper and lower bound of the objective function as the best and the worst value, respectively. It highlights the possible risk involved in the decision-making tool. A numerical example is presented to demonstrate the proposed solution procedure.

We consider the problem of determining the cumulative distribution function and/or moments of the optimal solution value of a nonlinear program dependent upon a single random variable. This problem is difficult computationally because one must in effect determine the optimal solution to an infinite number of nonlinear programs. Bereanu [Bereanu, B., G. Peeters. 1970. A ‘Wait-and-See’ problem in stochastic linear programming. An experimental computer code. Cashiers Centre Etudes Rech. Oper. 12 (3) 133–148.] has provided an algorithm to solve the distribution problem in the linear case based on extensions of the methods of parametric linear programming. (See also [Bereanu, B. 1967. On stochastic linear programming, distribution problems: stochastic technology matrix. Z. f. Wahrscheinlichkeitstheorie u. oerw. Gerbieter 8 148–152; Bereanu, B. 1971. The distribution problem in stochastic linear programming: the Cartesian integration method. Center of Mathematical Statistics of the Academy of RSR, Bucharest, 71–103 (mimeographed); Bereanu, B. 1970. Renewal processes and some stochastic programming problems in economics. SIAM J. Appl. Math. 19 308–322; Bereanu, B. 1973. The Cartesian integration method in stochastic linear programming. L. Collatz, W. Wetterlink, eds. Numerische Methoden bei Optimierungsaufgaben. Springer-Verlag Publishing Co., Inc., Basel; Prekopa, A. 1966. On the probability distribution of the optimum of a random linear program. SIAM J. Control 4 211–222.] for the analysis of more general linear programs.) This paper presents an extremely simple algorithm to solve the problem in the special case when all functions in the nonlinear program are homogeneous. In this instance the infinite class of optimal solutions are known linear homogeneous transformations of the optimal solution to a single nonlinear program. The distribution function may then be determined by substitution of an easily calculated variable into the distribution function of the random variable. The results are useful in the solution and analysis of a number of financial optimization problems. Problems from the analysis of optimal capital accumulation and portfolio separation are treated in some detail.

Using different dominance criteria in stochastic fuzzy linear multiobjective programming: A case of fuzzy weighted objective function

On stochastic programming I. Static linear programming under risk

Approximation in two-stage stochastic integer programming