Objective value

The solution to this program yields the following results Objective value 0.1168831E-02... 

TTie solution report from LINGO gives the following results Objective value 27373... 

In this section, the above mathematical model is applied to a literature example shown in Fig. 2.2 (Ierapetritou and Floudas, 1998). The SSN representation is given in Fig. 2.3b. Table 2.1 gives data for this example. 5 time points and a 12-h time horizon were used. Using less time points leads to a suboptimal solution with an objective value of 50, and using more time points than 5 did not improve the solution. It is worthy of note that, in this particular example, constraint (2.13) is redundant as mentioned earlier, since each unit is only performing one task. 

The results in the second and third columns were obtained using GAMS 2.5/OSL in a 600 MHz Pentium III processor, while those in the fourth and fifth columns were taken directly from Ierapetritou and Floudas (1998). The approach based on the SSN representation gives an objective value of 71.473 and requires only 15 binary variables, compared to 48 and 46 binary variables required in approaches proposed by Zhang, and Schilling and Pantelides, respectively. The formulation by Ierapetritou and Floudas (1998) initially consisted of 30 binary variables that were later reduced to 15 by exploiting one to one correspondence of units and tasks. It... 

Lastly, this chapter presents the concept of aggregation as a means of reducing the binary dimension in large-scale problems. In the examples cited, the objective values predicted by the aggregation model were very close to those predicted by the general formulation. However, the aggregation model requires a much smaller number of binary variables which is concomitant with significantly reduced computational effort. 

The overall model for this scenario involves 5614 constraints, 1132 continuous 280 binary variables. Three major iterations with an average of 1200 nodes in the branch and bound search tree were required in the solution. The objective value of 1560 kg, which corresponds to 33.89% reduction in freshwater requirement, was obtained in 60.24 CPU seconds. An equivalent of this scenario, without reusable water storage, i.e. scenario 2, resulted in 13% reduction in fresh water. Figure 4.12 shows the water recycle/reuse network corresponding to this solution. 

Table 4.4 is the summary of the mathematical model and the results obtained for the case study. The model for scenario 1 involves 637 constraints, 245 continuous and 42 binary variables. Seventy nodes were explored in the branch and bound algorithm. The model was solved in 1.61 CPU seconds, yielding an objective value (profit) of 1.61 million over the time horizon of interest, i.e. 6 h. This objective is concomitant with the production of 850 t of product and utilization of 210 t of freshwater. Ignoring any possibility for water reuse/recycle, whilst targeting the same product quantity would result in 390 t of freshwater utilization. Therefore, exploitation of water reuse/recycle opportunities results in more than 46% savings in freshwater utilization, in the absence of central reusable water storage. The water network to achieve the target is shown in Fig. 4.14. 

As shown in Table 4.4, the model for scenario 2, which is a nonconvex MINLP, consists of 1195 constraints, 352 continuous and 70 binary variables. An average of 151 nodes were explored in the branch and bound algorithm over the 3 major iterations between the MILP master problem and NLP subproblems. The problem was solved in 2.48 CPU seconds with an objective value of 1.67 million. Whilst the product quantity is the same as in scenario 1, i.e. 850 t, the water requirement is only 185 t, which corresponds to 52.56% reduction in freshwater requirement. The water network to achieve this target is shown in Fig. 4.15. 

More significantly, a suboptimal objective value of 2944.1 rcu was reported as an optimal solution. Using the uneven discretization of time formulation proposed in this chapter, a globally optimal value of 3081.8 rcu was obtained in 24.5 CPU s. Only 72 binary variables were necessary and the model solution was based on... 

The case study was solved using the uneven discretization of time formulation presented in this chapter. The mathematical model for the scenario without heat integration (standalone mode) involved 88 binary variables and gave an objective value of 1060 rcu. This value corresponds to the production of 14 t of product and external utility consumption of 12 energy units of steam and 20 energy units... 

In this section, the numerical solutions of the MINLP-model and of the MILP-model as presented in Sections 7.4 and 7.5 are compared with respect to their solution quality (measured by the objective values) and the required solution effort (measured by the computing time). In order to compare the MILP-solution with the MINLP-solution, the optimized values for the start times of polymerizations tn, the recipe assignments W, and the total holdups Mnr are inserted into the MINLP-model and the objective is calculated. To guarantee comparability of the results, the models were stated with identical initial conditions, namely t° = 0, = 2 Vk, pf = 0 Vs, and ra = 0.4 Vs (i.e., the variables defined at the beginning of the corresponding time axes are fixed to the indicated values). For the algorithmic solution procedure, all variables were initialized by 1 (i.e., the search for optimal values starts at values of 1 ), and none of the solvers was specifically customized. 

Gantt charts for the optimal solutions for the two instances are shown in Figures 8.7 and 8.8. The corresponding state number is shown within each rectangle. Model sizes, computational times and objective values are summarized in Table 8.3. The number of time intervals or points that was required in each case... 

The sequence of decisions obtained from the scheduler for all possible evolutions of the demand for the three periods is shown in Figure 9.4. The plant is started with an empty storage Mq = 0, no deficit from previous periods dp = 0, and the plant operation state on . The boxes contain the production decisions for each period while the circles contain the total objective after three periods for each scenario. The average objective value over the eight scenarios after three periods results as P = —16.05. The small figures on the right provide the evolution of the storage Mf, the deficit Bf, the late sale Mf, the sale Mi, the production x , and the objective p per period for each scenario. 

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 ... 

Fig. 9.15 ES evaluation best objective value vs. CPU-time (best lower bound = —20.6).

Table 10.2 Test results. The values refer to the points of time at which the first-best solution was found, the total effort to terminate was nearly the same (5 10 nodes). Times are in seconds, initial quantities of S0 are in tons, objective values (mspan) are in time units and the overproduction (op) is measured in tons.

Due to the campaign structure, the existing decomposition techniques in the SNP optimizer like time decomposition and product decomposition are not applicable. For problems with this structure it is possible to use the resource decomposition in case a good sequence of planning of the campaign resources can be derived. However, in our case, problem instances could be solved without decomposition on a Pentium IV with 2 GHz in one hour to a solution quality of which the objective value deviates at most one percent from the optimal objective function value. 

In addition to providing optimal x values, both simplex and barrier solvers provide values of dual variables or Lagrange multipliers for each constraint. We discuss Lagrange multipliers at some length in Chapter 8, and the conclusions reached there, valid for nonlinear problems, must hold for linear programs as well. In Chapter 8 we show that the dual variable for a constraint is equal to the derivative of the optimal objective value with respect to the constraint limit or right-hand side. We illustrate this with examples in Section 7.8. 

These formulas agree with the previous results for b = 1. The minimal objective value, sometimes called the optimal value function, is... 

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