Stochastic Programming Models

The models discussed above assume that all parameters are known with certainty, which is not the case in real-life situations. To obtain robust results, the impact of uncertainty needs to be assessed. Stochastic programming is one of the techniques allowing accounting for stochastic parameters. 

Many of the stochastic programming models developed for supply chain configuration have demand as a stochastic parameter. Demand uncertainty usually is represented by multiple demand scenarios (Mirhassani et al. 2000 Tsiakis et al. 2001). In this case, a prototype objective function can be expressed as 

Other stochastic parameters can also be represented by evaluation of multiple scenarios (e.g., Gutierrez et al. 1996). The obvious limitation of this approach is a limited number of considered scenarios and there is little assurance that the coverage of uncertainty has been adequate. 

Kim et al. (2002) develop a model for determining ordering quantities from suppliers for a fixed supply chain network subject to demand uncertainty. The demand uncertainty is represented using demand probabdity density function and an iterative model-solving procedure is developed without relying on using scenarios. 

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A two-stage stochastic programming model for a 3-tier reverse logistics network design problem with stochastic demand and supply [7]... 

Contrary to t3q>ically deterministic mathematical models available in the literamre, real-world apphcations are usually surrounded with uncertainty. The two main approaches for dealing with uncertainty are stochastic programming and robust optimization. For developing stochastic programming models, probability distributions of uncertain parameters should be known in advance. However, in many practical situations, there is no information or enough information for obtaining probability distribution of uncertain parameters. Robust optimization models are viable answers to these situations via providing solutions that are always... 

Probability theory and mathematical statistics had long been considered the only method of dealing with indetermination until the 1960s although in reality there were many undefined parameters. Dealing with optimization problems, one would often find the so-called stochastic programming model formed by those parameters that appeared in the model as random variables [1]. 

It is generally believed that study of the stochastic programming model began in 1955 when Beale [2] and Dantzig [3] respectively proposed the two-stage stochastic programming model. In 1959, Chames and Cooper [4] advanced a chance-con-strained programming model. 

Taking into account that decision-makers do not always care about maximizing revenue, but how to achieve the optimal revenue in the sense of probability, we apply stochastic chance-constrained programming theory to translate the model into the stochastic programming model under chance constraints so that the optimal decision objective with a certain confidence level can be expressed. 

Bozotgi-Amiri A, Jabalameli MS, Mirzapour AI-e-Hashem SMJ (2013) A multi-objective robust stochastic programming model for disaster relief logistics under uncertainty. OR Spectmm. doi 10.1007/s00291-011-0268-x... 

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