Stochastic Chance-Constrained Programming

In fact, in the actual market environment, the supply chain node enterprises may face many kinds of real risks, under these circumstances, the decision-makers need to consider how to maximize supply chain benefits in terms of probabUily. Stochastic chance-constrained programming theory has a practical significance in dealing with such issues. Following is the brief introduction of stochastic chance-constrained programming theory. 

Stochastic chance-constrained programming is proposed by Charnes and Cooper [3] in 1959, which is an optimization theory in terms of probability. It is mainly for constraint conditions including random variables and the decisions must be made before random variables are observed. A principle is adopted with consideration that the decisions are made in the event of adverse situations which may not satisfy the constraints decisions are allowed not to meet the constraints in some degree, but the probability of constraints being satisfied should be kept not less than a confidence level a [1, 2]. 

Consider a mathematical programming model with random variables  

This type of chance constraints is named as joint chance constraints. Where Pr - represents the probability of event established in , a point x is feasible if and only if the probability of event gj x, ) 0,j = 1,2./ is no less than a, which means the probability of constraint violation is less than (1 — a). 

After the presentation of chance-constrained programming, many researchers studied it. According to literature [1, 2], in stochastic environment, if decision makers want to maximize the optimistic value of the objective function, the following Chance-Constrained Programming Model (CCPM) can be established Assume / = max / Pr /(x, ) / /,  

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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. 

As the number of node enterprises which constitute the supply chain can be infinitely expanded, to solve the stochastic chance-constrained programming model, a computational layer is to be divided from the stmcture of supply chain in planning period T. Then the layered computation is adopted on demand. Firstly the overall strncmre of the supply chain should be determined in planning period T before computation. Secondly, the upstream and the downstream enterprises of the supply... 

In order to further test the validity of the algorithm and the stochastic chance-constrained programming model, we start the simulation process by taking different values of parameters of the hybrid intelligent algorithm. 

Optimization of decentralized control supply chain logistics planning under uncertain environment is studied in the book. On the basis of conception of supply chain cell, the uncertainty of price factor of up/downstream materials of nodal enterprises is taken into consideration and stochastic chance constrained programming model of integrated logistics planning for decentralized control... 

Appendix A Two-Stage Stochastic Programming 183 Appendix B Chance Constrained Programming 185 Appendix C SAA Optimal Solution Bounding 187... 

At last we designed a numerical example to test the validity of the model by simulation and then analyzed the sensitivity of the parameters in the model. The simulation results indicate that the optimization problem of decentralized control supply chain planning for the supply and demand prices between node enterprises under stochastic environment can be effectively solved by the chance-constrained programming model we established and the algorithm we designed. 

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