Multiperiodicity

Papalexandri, K. P., and Pistikopoulos, E. N. (1994). A multiperiod MINLP model for the synthesis of heat and mass exchange networks. Comput. Chem. Eng. 18(12), 1125-1139. 

Iyer RR and Grossmann IE (1998) Synthesis and Operational Planning of Utility Systems for Multiperiod Operation, Comp Chem Eng, 22 979. 

Multiperiodic SPB stars were found in the whole range of the V magnitudes in the searched area. Unlike in the Galaxy, these stars populate the region of early B or even late O-type stars. 

Four multiperiodic [3 Cephei and eleven SPB stars were found in the SMC. There are also two monoperiodic short-period variables that are good candidates for / Cephei stars in this galaxy. 

Many multiperiodic variables with periods in the range between 0.25 and 1.5 d were found. They can be identified with the known emission-line objects (Be stars), so that their light variations are of A Eri-type. 

Porkka, P., Vepsalainen, A.P.J. and Kuula, M. (2003) Multiperiod production planning carrying over setup time. Int. f. Product. Res., 41, 1133-1148. 

Bhatia, T. K. and L. T. Biegler. Multiperiod Design and Planning with Interior Point Methods. Comp Chem Engin 23 919 (1999). 

Planning Economic Steady state, single or multiperiod, discrete-event, material flows... 

Fisher M, Ramdas K, Zheng Y (2001) Ending Inventory Valuation in Multiperiod Production Scheduling. Management Science 45 679-692... 

Sahinidis, N.V. and Grossmann, I.E. (1991a) Reformulation of multiperiod MILP models for planning and scheduling of chemical processes. Computers ei Chemical Engineering, 15, 255. 

Neiro, S.M.S. and Pinto, J.M. (2005) Multiperiod optimization for production planning of petroleum refineries. Chemical Engineering Communications, 192, 62. 

We get a very similar time behavior in subharmonic generation where/1 = 0 and /2 / 0. Self-pulsation and multiperiodic evolution of intensities have been found. However, these findings are not investigated here. 

A more complicated behavior of the system (3) is manifested if the time-dependent driving field and damping are taken into account. Let us assume that the driving amplitude has the form /1 (x) =/o(l + sin (Hr)), meaning that the external pump amplitude is modulated with the frequency around /0. Moreover,/) = 0 and Ai = 2 = 0. It is obvious that if we now examine Eq. (3), the situation in the phase space changes sharply. In our system there are two competitive oscillations. The first belongs to the multiperiodic evolution mentioned in Section n.D, and the second is generated by the modulated external pump field. Consequently, we observe a rich variety of nonlinear oscillations in the SHG process. 

Fig. 16a symmetric limit cycles for the second-harmonic mode (GCL) and in Fig. 16b, an nonsymmetric phase portrait example for 7) = 0.5 for BCL. In both cases the phase point settles down into a closed-loop trajectory, although not earlier than about x > 200. An intricate limit cycle is usually related to multiperiod oscillations. For example, the cycle in Fig. 16a corresponds to five-period oscillations of the fundamental and SHG modes intensity, and the phase portrait in Fig. 16b resembles the four-period oscillations (see Fig. 17). Generally, for 7) > 0.5, we observe many different multiperiod (even 12-period) oscillations in intensity and a rich variety of phase portraits.

It is interesting that envelope functions can also behave as multiperiod oscillations. This takes place if we take into account small damping. By way of an example, for the damping constant yj = y2 = 0.1, the envelope function has a feature of two period doubling oscillations. 

The multiperiod synthesis-analysis-resynthesis algorithm is actually decomposed into two stages (Floudas and Grossmann, 1987b) ... 

Step 3. Formulate and solve the multiperiod MILP transshipment model of Floudas and Grossmann (1986) to determine a minimum set of stream matches for feasible operation in all the periods and the heat transferred in each match in each period. 

Step 5. Derive the multiperiod superstructure based upon the matches and heat transferred in each match predicted by the multiperiod MILP transshipment model. Formulate an NLP to optimize the superstructure (Floudas and Grossmann, 1987a) to give the HEN structure and exchanger sizes which minimize investment cost. 

Al) Energy balances on each hot process and utility stream in each temperature interval (TI) of the multiperiod MILP transshipment model. Each energy balance involves the residuals (heat cascaded) to and from the TI and the heat transferred in each stream match in the TI. 

By formulating and solving the multiperiod MILP transshipment model for these two periods of operation, the following set of matches is identified ... 

In addition, both the FI target-based synthesis procedure and the multiperiod synthesis procedure suffer from the fact that they synthesize a HEN for a specified uncertainty range and cannot directly consider the trade-off between resilience and total HEN cost. At present, the best way of evaluating this trade-off with these methods is to synthesize HENs for several different sizes of the uncertainty range and then to compare the HENs. The downstream path method (combined with sensitivity tables) evaluates this trade-off much more easily and directly (subject to the quantitative limitations discussed above). 

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