Abstract
We study a class of stochastic bi-criteria optimization problems with one quadratic and one linear objective functions and some linear inequality constraints. A hybrid method of chance-constrained programming (CCP) combined with variance expectation (VE) is proposed to find the optimal solution of the original problem. By introducing the expectation level, the bi-criteria problem is converted into a single-objective problem. By introducing the confidence level and the preference level of decision maker, we obtain a relaxed robust deterministic formulation of the stochastic problem. Then, an interactive algorithm is developed to solve the obtained deterministic model with three parameters, reflecting the preferences of decision maker. Numerical experiments show that the proposed method is superior to the existing methods. The optimal solution obtained by our method has less violation of the constraints and reflects the satisfaction degree of decision-maker.
Highlights
In many fields of industrial engineering and management sciences, there often exist some uncertainties, such as the return rate of security and the amounts of demand and supply
For the first stochastic inequality in 2.1, we introduce the so-called chance constraint method to convert it into a deterministic inequality constraint, which is used to guarantee that the stochastic constraint is satisfied with a probability as higher as possible.For the general stochastic constraints A w x ≥ b w, we obtain their deterministic formulations by expectation method as done in 9
From Model 2.9, we are going to develop an interactive algorithm to obtain an optimal solution of the original problem 1.1 such that there is less violation of constraints
Summary
In many fields of industrial engineering and management sciences, there often exist some uncertainties, such as the return rate of security and the amounts of demand and supply. Many attentions have been paid to construct optimization models with uncertain parameters for the decision problems in the field of management science and to design some efficient solution methods for these optimization models. For this connection, one can see 1–9 and the references therein. The chance constraints approach can guarantee that the obtained solution has less degree of constraint violation see 4, 11 Based on such a reformulation for the original problem, an interactive algorithm will be developed to find its solution with some satisfaction degree.
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