Stochastic Expected Value Programming

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]. 

A basic assumption of stochastic programming is that the probability distribution of the random variable is known. The target then is to find an optimal solution that makes the expected value of the system to be minimum (or maximum). According to the type of the objective function and constraints, the stochastic programming problem can be divided into stochastic linear programming problems and stochastic nonlinear programming problems. 

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. 

This book analyses uncertainty of market demand and price of the finished products and builds the model mainly based on stochastic programming theory [5-9]. In the following the stochastic programming theory will be presented and discussed. 

---

Li YF, Xia GP, Yang YX et al (2005) Global supply chain tactical planning model based on fuzzy stochastic expected value programming. Syst Eng Theory Pract 8 1-9... 

The first part briefly introduces the stochastic expected value programming theory. With the discussing on the expected value model which is a convex programming, Theorem 4.2 is put forward and proved. Furthermore we get the conclusion that if the expected value model is a convex programming and there exists an optimal solution, then any local optimal solution will be the global optimal solution. 

Based on the hypothesis that pfio t) and w so(t) are random variables independent mutually, the stochastic expected value programming model for supply chain logistics planning, which aims at the maximum profit of the entire supply chain in planning period T is ... 

For the stochastic expected value programming model in supply chain logistics planning, it is assumed that the external market demand and market price are random parameters while the other parameters within the system are constants at every stage in the planning period. 

Objective function (6.8) and its restrictions might not have strict mathematical sense as the models includes fuzzy random variables. We transform the model by using the theory in Sect. 6.1 to the following fuzzy stochastic expected value programming model. 

---