Multi-period model

A MILP model of the aggregated scheduling problem of the EPS process was proposed by Sand and Engell [16]. The model is formulated as a discrete time multi-period model where each period i e 1,..., 1 corresponds to two days. The degrees of freedom of the aggregated problem are the following discrete production decisions ... 

One weakness or concern in having a multi-period model is that profit and the other criteria values are considered only for a specific number of periods, and that the decisions to be made depend on the demands of each period. To take care of this, a demand sensitivity analysis should be used, such as a... 

In this fashion, we extend our deterministic model with a prediction horizon of H = 2 to a multi-stage model. The multi-stage tree of the possible outcomes of the demand within this horizon (starting from period i = 1) with four scenarios is shown in Figure 9.5. Each scenario represents the combination fc out of the set of all combinations of the demand outcomes within the horizon. The production decision x has to be taken under uncertainty in all future demands. The decision xj can react to each of the two outcomes of d i, but has to be taken under uncertainty in the demand di. The corrective decisions are explicitly modeled by replacing xj by two variables 2,1 and 2.2 ... 

Concluding, the areas mostly impacted by the global scope such as global material flow planning and multi-period transport and transit inventory planning are covered best by global models (s. table 15). 

Sahinidis, N.V. and Grossmann, I.E. (1992) Reformulation of multi-period MILP model for planning and scheduling of chemical processes. Operations Research,... 

Capacity. Both uncapacitated (-) and capacitated (C) models exist. Some of the multi-period, capacitated models also allow expansions (E), relocations (RL) or reductions (R) of capacity throughout the planning horizon. 

Fleischmann et al. (2006) provide a global production network planning model used at BMW that extends the simpler load planning model proposed by Flenrich (2002). The model is a multi-period, multi-product model with an objective function that maximizes the pre-tax net present value of the network. It includes decisions on product-plant allocation, production volumes, material sourcing volumes by supply region, structural and product-specific investments and use of overtime capacity. A major contribution of the model is the incorporation of the time-distribution of investment expenditures typically observed in automobile production networks. While tariffs are included in the transportation costs, the model does not consider further aspects of international trade such as currencies, duty drawbacks or local content rules which play a major role in practice. 

Melo et al. (2005) propose a multi-period, deterministic, multiple-product MILP model for strategic supply chain planning. The model does not impose any restrictions on the number and type of facilities and the transportation links between facilities. The basic model explicitly covers relocation of capacity to new facilities. It can be extended to include capacity expansions and reductions. To this end, two fictitious, non-selectable facilities are introduced that provide additional or absorb excessive capacities. Capacity is assumed to be adjustable on a continuous scale but an extension to modular capacity is also provided. The model is very... 

Kouvelis et al. (2004) present a relatively simple multi-period MILP plant location model for global production network design with investment decisions only allowed in the first period. The production system consists of component-dedicated manufacturing sites and final assembly sites. It is limited to two production levels and one final product. The objective function maximizes the NPV of the production network. The main purpose of the model is to analyze the effects financing subsidies, tax regimes, tariff structures and local content requirements have on optimal network design. The analysis is based on theoretical considerations and a numerical example. More complex aspects of international trade such as duty drawbacks are not considered. 

The model proposed by Bhutta et al. (2003) is a multi-period, deterministic multiple-product MILP model integrating plant location, production, distribution and investment planning in a global environment. It is relatively simple both mathematically (no binary decision variables but integer production quantities) and with respect to the assumptions made for key modeling parameters. Capacity can be modified continuously without lower or upper bounds. International features are limited to exchange rates and tariffs. 

Antunes A, Peeters D (2001) On solving complex multi-period location models using simulated annealing. European Journal of Operational Research 130 190-201... 

Model Fitting to Mixing-Cell Data Multiple-site kinetic models have been used to describe pesticide and herbicide movement in soils (15,16,17), cesium migration in columns (18), and strontium migration in a sandy aquifer over a twenty-year time period (1L). The results of the selective extraction procedures in all experiments discussed here suggest that a multi-site model should provide a better fit of the data than a single-site model. This hypothesis is supported by the variances in Table I, with the possible exception of selenium. 

The remainder of this paper is organized as follows Section 2 relates our model to well known approaches based on multi-period investment models proposed in the literature. In Section 3, our investment planning model is introduced while in Section 4 the investment planning strategy of a generic process is presented as a case-study. Section 5 gives a discussion on the results of the case study. Finally, in Section 6 our findings are summarized. 

The proposed optimization approach falls into the category of multi-period investment models. Various related approaches can be found in the literature, (e.g. [2], [3]), including MILP optimization for long range investment planning. Some authors also consider stochastic elements to cover the potential uncertainties, which, however, is beyond the scope of this paper. 

A novel stochastic multi-period design/planning/scheduling MILP model of a multiechelon SC with financial considerations is used as a predictive model in this work. The model assumes that different technological equipment is available to be installed in potential sites and assists in their selection. Furthermore, the model allows the expansion of plant equipment capacities, not only in the first planning period. Regarding the financial area, the mathematical program endeavors to evaluate the shareholder value. 

FIGURE 4-32 Diurnal cycle of Vd for ozone for a deciduous forest, averaged over the period July 7 to August 30, 1988. Observed values of Vd and values estimated by four models are shown. (Reprinted from Atmospheric Environment, Vol. 30, L. Zhang, J. Padro, and J. L. Walmsley, A Multi-Layer Model vs. Single-Layer Models and Observed 03 Dry Deposition Velocities, pp. 339-345, Copyright 1996, with permission from Elsevier Science.)... 

When the results in Fig. III.29 B, D, E, F are compared with the generic multi periodic limit cycle in Fig. IV.4 the relationship between the model solutions and the generic possibilities can be easily seen. 

The limit bundle shown in the abstract model (1976-2) is also generically a multi periodic limit cycle. In the abstract model (1974-4) linked limit cycles are also observed. These linked limit cycles are examples of the generic multi limit cycle shown in Fig. IV.4. 

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