Objectives stochasticity

Multi-objective linear programming (MOLP) Multi-objective stochastic integer linear programming Interactive MOLP Mixed 0-1 MOLP... 

Multi-objective, stochastic, and nonlinear mathematical programming models are other models that find application in supply chain configuration. 

Franca, R.B., Jones, E.C., Richards, C.N. and Carlson, J.P. 2010. Multi-objective stochastic supply chain modeling to evaluate tradeoffs between profit and quality. International Journal of Production Economics, 127, 292-299. 

Corrosion likelihood describes the expected corrosion rates or the expected extent of corrosion effects over a planned useful life [14]. Accurate predictions of corrosion rates are not possible, due to the incomplete knowledge of the parameters of the system and, most of all, to the stochastic nature of local corrosion. Figure 4-3 gives schematic information on the different states of corrosion of extended objects (e.g., buried pipelines) according to the concepts in Ref. 15. The arrows represent the current densities of the anode and cathode partial reactions at a particular instant. It must be assumed that two narrowly separated arrows interchange with each other periodically in such a way that they exist at both fracture locations for the same amount of time. The result is a continuous corrosion attack along the surface. 

Perikinetic motion of small particles (known as colloids ) in a liquid is easily observed under the optical microscope or in a shaft of sunlight through a dusty room - the particles moving in a somewhat jerky and chaotic manner known as the random walk caused by particle bombardment by the fluid molecules reflecting their thermal energy. Einstein propounded the essential physics of perikinetic or Brownian motion (Furth, 1956). Brownian motion is stochastic in the sense that any earlier movements do not affect each successive displacement. This is thus a type of Markov process and the trajectory is an archetypal fractal object of dimension 2 (Mandlebroot, 1982). 

Transforms are important in signal processing. An important objective of signal processing is to improve the signal-to-noise ratio of a signal. This can be done in the time domain and in the frequency domain. Signals are composed of a deterministic part, which carries the chemical information and a stochastic or random part which is caused by deficiencies of the instmmentation, e.g. shot noise... 

Stochastic methods do not need auxiliary information, such as derivatives, in order to progress. They only require an objective function for the search. This means that stochastic methods can handle problems in which the calculation of the derivatives would be complex and cause deterministic methods to fail. 

Having evaluated the system performance for each setting of the six variables, the variables are optimized simultaneously in a multidimensional optimization, using for example SQP, to maximize or minimize an objective function evaluated at each setting of the variables. However, in practice, many models tend to be nonlinear and hence a stochastic method can be more effective. 

Various search strategies can be used to locate the optimum. Indirect search strategies do not use information on gradients, whereas direct search strategies require this information. These methods always seek to improve the objective function in each step in a search. On the other hand, stochastic search methods, such as simulated annealing and genetic algorithms, allow some deterioration... 

In summary, models can be classified in general into deterministic, which describe the system as cause/effect relationships and stochastic, which incorporate the concept of risk, probability or other measures of uncertainty. Deterministic and stochastic models may be developed from observation, semi-empirical approaches, and theoretical approaches. In developing a model, scientists attempt to reach an optimal compromise among the above approaches, given the level of detail justified by both the data availability and the study objectives. Deterministic model formulations can be further classified into simulation models which employ a well accepted empirical equation, that is forced via calibration coefficients, to describe a system and analytic models in which the derived equation describes the physics/chemistry of a system. 

The sequence of decisions obtained from the stochastic scheduler for all possible evolutions of the demand for the three periods is provided in Figure 9.7. The sequence of decisions obtained by the stochastic scheduler differs from that obtained by the deterministic one, e.g., xi(ti) = lOinstead ofxi(ti) = 6. The average objective for the stochastic scheduler after three periods is P = —17.65. 

The performance ofthe deterministic and of the stochastic scheduler is compared in Figure 9.8. The figure shows the objective for all scenarios and the average objective. The stochastic scheduler improves the average objective by approximately 10% and for five out of eight scenarios. On the other hand, the stochastic scheduler produces a larger variation in the objective of the scenarios. 

Fig. 9.7 Stochastic scheduler sequence of decisions and results for all scenarios (average objective after three periods P = —17.65), compare to Figure 9.4.

Fig. 9.8 Deterministic vs. stochastic scheduler comparison of the objective after three periods.

Plugging the first-stage solution of the EV problem xEV into the stochastic program (2S-MILP) gives the expected result of using the EV solution (EEV problem). The solution of the EEV problem is not necessarily optimal for the original 2S-MILP. Consequently, the optimal objective value of the EEV problem is always greater than (or at least equal to) the optimal objective value of the 2S-MILP, such that the objective of EEV is an upper bound for the optimal solution of the 2S-MILP ... 

The advantage of using a 2S-MILP instead of the corresponding deterministic approach is measured by the value of the stochastic solution (VSS) which is the difference of the respective optimal objective values ... 

An uncertainty conscious scheduling approach for real-time scheduling was presented in this chapter. The approach is based on a moving horizon scheme where in each time period a two-stage stochastic program is solved. For the investigated example it was found that the stochastic scheduler improved the objective on average by 10% compared to a deterministic scheduler. 

Robust planning is a specific research area within Operations Research (Scholl 2001). Generally, robustness can be defined as the insensitivity of an object or system against (stochastic) external influences (SchneeweiB 1992 reviewed by Scholl 2001, p. 93). A plan is robust, if the realization of the plan - also in a slightly modified form - leads to good and/or accept-... 

In the considered value chain planning problem, the uncertainty of spot sales prices impacts the profitability of the overall value chain plan, since volume decisions can lead to profit-suboptimal plans, if the average sales price cannot be realized as planned. Therefore, price volatility is considered as an external (stochastic) influence in the considered value chain planning problem. The following model extensions account for this uncertainty and try to derive methods to achieve more robust plans with respect to profit results with contributions from Habla (2006). The objective of the proposed modeling approach is to maximize profit for the entire value chain network. It is assumed that the company behaves risk-averse in face of the price uncertainty. 

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