Essay Example: Bi-level Mathematical Programming and Fuzzy Programming Approach

A hierarchical administrative structure is necessary for a planning problem for making decisions that involve conflicting and independent objectives of individuals. An example of mathematical programming that uses two levels of decision-making for decentralized planning problems is Bi-level mathematical programming (BLMP). BLMP works such that a follower looks for a suitable decision following the decision made by their leader. (Bard 1998; Dempe 2003 Colson et al., 2007). BLMP is implemented in electricity markets (Zhang et al., 2011), principle-agent problems (Cecchini et al. 2013), and transport network design (Gzara, 2013; Fontaine, and Minner, 2014) and they use transformation and vertex enumeration concepts. The Kth–best solution is an example of a vertex enumeration method pioneered by Bialas and Karwarn (1982, 1984). An increasing number of variables in the process increases storage requirements, although it aims to find a globally optimal solution using the simplex method. The grid search algorithm creates a series of equalities from product terms, as established by Bard and Falk (1982).

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Chen et al. (1995) studied the use of the marginal value function in the parametric lower-level problem to show the optimality conditions and the regularity of the BLP problem. Falk et al. (1995) studied the BLP problem by converting it into an equivalent non-differentiable optimization problem using local optimizers of the lower-level problem with strong stability. Gendreau et al. (1996) proposed the use of an adaptive search method to solve the NP-hard problem as the BLP problem is in the linear hierarchical decision process as an instance. The use of fuzzy triangular numbers was introduced to address the BLP problem by Safai et al. (2014). It involves bounding variable constraints of three deterministic L.P. problems. Another approach to solving a class of fully fuzzy BLP was proposed by Ren (2016) to use a ranking function and deviation degree measures. Finding a fuzzy optimal solution to the FBLP problem using an interactive programming method created by Ren (2015). A bi-level multi-objective optimization model for solving the evacuation location assignment was presented by Hammad (2019). Using a comprehensive algorithm framework to solve mixed-integer problems with a cut approach and generalized branch was created by Tahernejad et al. (2020). The theory of stochastic variations and methods is part of the stochastic programming deals (Segupta, 1972). It involves two-stage programming and chance-constrained programming.

Probabilistic programming interactive methods were introduced by Leclercq (1982) and Teghem et al. (1986). The distribution of right-hand side constraints in multi-objective probabilistic L.P. was studied by Sinha et al. (1998). Data is useful for setting up a model in areas like operators' research and system analysis. Zadeh (1965) introduced the fuzzy set theory. Dubois and Prade (1980) studied fuzzy numbers in fuzzy linear constraints. The fuzzy set theory is useful in managing risk in finance as uncertain and imprecise elements are used in the decision. Fuzzy quantities and constraints may represent uncertainty in financial markets and the returns on assets. Bellman and Zadeh (1970) developed the use of a fuzzy environment for decision-making. Hanan (1981), Narasimhan (1980), and Zimmerman (1978) were the first to feature goal programming problems. Solving the multi-criteria optimization problem using the min-operator was applied by Leberling (1981). Zimmerman (1985) studied the use of multi-objective functions in the Fuzzy L.P. Sinha and Biswal (2000) show the features of a bi-level organization as follows:

• The set of available decision space and objective function reflect the decision maker's problem's external effect.
• Externalities as units of action affect the benefits of a group as it maximizes or minimizes their benefits.
• A decision is executed from the upper to the lower level sequentially.
• The hierarchical structure has interactive D.M. units.

When the constraints on the right-hand side follow a normal joint distribution, and they are standard random variables, the bi-level probabilistic L.P. is introduced in this paper. The optimal compromise solution of the stochastic bi-level L.P. problem is deduced by a fuzzy programming approach in the conversion of the probabilistic problem into an equivalent deterministic problem as the first step. The probabilistic BLP problem is formulated in section, and the optimal compromise solution of the SBLP problem is obtained in section 3 with the application of the fuzzy programming approach. There is an illustration of the proposed solution procedure by a numerical example in section 4, while section 5 features a report on the concluding remarks.

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