Programming chance constrained

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

Charnes, A. and Cooper, W. (1959) Chance-constrained programming. Management Science, 6, 73. 

Chames A. and Cooper W.W. 1959. Chance-constrained programming, Manag. Sci., 6, 73. Chames A., Cooper W.W. and Miller M.H. 1959. Application of linear programming to financial budgeting and the costing of funds. In, The Management of Corporate Capital, Solomon E. (Ed.). Free Press of Glencoe, Illinois. 

Bilsel, R. U. and A. Ravindran. 2011. A Multi-objective chance constrained programming model for supplier selection. Transportation Research Part B. 45(8) 1284-1300. 

Liu [32] divided fuzzy programming into fuzzy expected value programming, fuzzy chance-constrained programming and fuzzy relevant-chance programming and so on. Specific approach is also connected with calculating method of fuzzy number, e.g. the comparative method, the calculation of the mean etc. [33]. 

Zhou et al. [61] described production planning of multi-location plant and distributors on condition that unit production cost, production capacity and demand are fuzzy parameters. They built up a fuzzy expected value model and fuzzy related chance-constrained programming model in consideration of different decision criteria and discussed a clear equivalent form of the fuzzy programming model when... 

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]. 

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, ) >/ > /, ... 

The above model is called a Maxi-max chance-constrained programming model, max/ is equivalent to max maxj/. a and / represent the objective function and the confidence level of constraints whose values are given in advance respectively, max/ is the maximum value of objective function/(x, ) with a confidence level at /. 

Under normal circumstances, chance-constrained programming is not necessarily convex, even assuming that aU functions gj x, ),j = 1,2,..., p are always convex. It still cannot guarantee the convexity of the chance constraints set, unless certain conditions are satisfied. These conditions may refer to that gj x, ),j= 1,2,..., p is united convex on (x, and probability Pr is pseudo-concave. In fact, it is... 

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. 

Due to the fact that the model consisting of objective function (5.17) and constraints contains random variables makes the max and the constraints have no complete mathematical sense, we need to transform the model accordingly. Considering the reliability of the actual decision-making environment and the uncertainty of parameters, this book wiU use the chance-constrained programming theory to... 

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. 

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. 

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... 

Bilsel, R. Ufuk, and Ravi Ravindran. A Multi-Objective Chance Constrained Programming Model for Supplier Selection imder Uncertainty. Transportation Research Part B 45, no. 8 (2011) 1284-1300. 

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