### Uncertain random programming models for fixed charge multi-item solid transportation problem

Hasan Dalman
2018 New Trends in Mathematical Sciences
This paper presents uncertain random programming models for the fixed charge multiobjective multi-item solid transportation problem, which contains uncertain random variables for fixed charges, unit transportation costs, source, destination and conveyance constraints. Utilizing both uncertainty and randomness, the uncertain random programming model is first turned into an expected value programming model under chance constraints. Thus a deterministic model of the uncertain random model is
more » ... ndom model is obtained. Finally, numerical examples are given to illustrate the models. 38 H. Dalman: Uncertain random programming models for fixed charge multi-item solid transportation problem Since then, the uncertainty theory becomes a branch of mathematics. Also, uncertainty theory have studied both theoretically and practically in literature [10,18]. TPs based on the uncertainty theory has been studied by some scholars. Cui and Sheng [19] gave an uncertain programming model for the STP. Zhang et al.[20] investigated uncertain fixed charge solid transportation problem, and Gao et al. [21] presented uncertain models on railway transportation planning problem. Dalman [22] introduced a multi-item STP with uncertain variables. However, randomness and uncertainty usually consist complex systems. After the uncertain programming is presented in [23], Liu [24] introduced chance theory for representing such types of complex events including both uncertainty and randomness. Then some basic notations were given. After that, Liu [25] showed the operational law for calculating a monotone function of uncertain random variables and introduced the formula to calculate expected value. Uncertain random programming models have been investigated in literature [26]-[30] and so on. According to my reading, no work has been given on uncertain random programming model for fixed charge multi-item solid transportation problems with uncertain random variables. Thus, this paper focuses on uncertain random programming for fixed charge multi-item solid transportation problem with uncertain random variables. Using the expected value of each objective function under the chance constraints, the model is transformed into a deterministic model. Finally, numerical examples are presented to illustrate the models. This paper is constructed as follows: Section 2 presents some definitions and theorems about uncertainty and chance theory. Section 3 presents a definition for an uncertain random programming model. Based on uncertainty and randomness, the fixed charge multi-item STP model is modeled in Section 4. Numerical examples are given to illustrate different fixed charge multi-item STP model with some uncertain random variables in Section 5. Preliminaries Uncertainty theory Basic definitions and notations of uncertainty theory are given here. Definition 1. Let L be a σ -algebra on a nonempty set Γ . A set function M is called an uncertain measure if it satisfies the following axioms: Axiom 1. (Normality Axiom) M{Γ } = 1; Axiom 2. (Duality Axiom) M{Λ } + M{Λ c } = 1 for any Λ ∈ L; Axiom 3. (Subadditivity Axiom) For every countable sequence of {Λ i } ∈ L, we have