Maximum Profit Problem

Kerkhof and Vissers and Mujtaba (1989) considered the operating cost Cj as constant, but in practice it may vary, say, with different boilup rates. More general expressions (Logsdon et al., 1990, Mujtaba and Macchietto, 1996) could also be used to evaluate the operating cost per hour. When C = 1.0, C2 = C3 = ts = 0, the maximum profit problem results in a maximum productivity problem (Mujtaba, 1989, 1999). 

Application to Batch Distillation Maximum Profit Problem Kerkhof and Vissers (1978) considered the following profit function ... 

Referring to the simple zero holdup model (described in 5.5.1.1), the maximum profit problem can now be written as ... 

Table 5.6 presents the results of a set of 5 maximum profit problems considered by Kerkhof and Vissers (1978). Each problem has different column configuration (i.e. different number of plates, N), handles different feed mixture (characterised by different relative volatility, a)y has different distillate product quality (xq), has different cost values (i.e. Pr, C0) and different set up times (ts). The initial charge, B0 = 100 kmol, V= 60 kmol/hr, t,= 1 hr, and Cf= 150 /hr. All cases deal with binary mixtures of different initial composition, xB0. 

Note that for a fixed operation time, t in Equation 9.1, the profit will increase with the increase in the distillate amount and a maximum profit optimisation problem will translate into a maximum distillate optimisation problem (Mujtaba and Macchietto, 1993 Diwekar, 1992). However, for any reaction scheme (some presented in Table 9.1) where one of the reaction products is the lightest in the mixture (and therefore suitable for distillation) the maximum conversion of the limiting reactant will always produce the highest achievable amount of distillate for a given purity and vice versa. This is true for reversible or irreversible reaction scheme and is already explained in the introduction section. Note for batch reactive distillation the maximum conversion problem and the maximum distillate problem can be interchangeably used in the maximum profit problem for fixed batch time. For non-reactive distillation system, of course, the maximum distillate problem has to be solved. 

Mujtaba and Macchietto (1997) proposed a new and an alternative technique that permits very efficient solution of the maximum profit problem using the solutions of the maximum conversion problem already calculated. This is detailed and explained in the following using again the ethanol esterification example presented in the previous section. 

The dynamic optimisation problem P2 now results in a single variable algebraic optimisation problem. The only variable to be optimised is the batch time t. The solution of the problem does no longer require full integration of the model equations. This method will solve the maximum profit problem very cheaply under frequently changing market prices of (CD/, CB0, C ) and will thus determine new optimum batch time for the plant. The optimal values of C, Dh r, QR, etc. can now be determined using the functions represented by Equations 9.2-9.5. 

For a given product purity of x D = 0.70, Mujtaba and Macchietto (1997) solved the maximum profit problem for a number of cost parameters using the method described above. The results are presented in Table 9.3. For each case, Table 9.3 also shows the optimal batch time, amount of product, reflux ratio, total reboiler duty and maximum conversion (calculated using the polynomial equations). 

Table 9.3. Summary of the Results for Maximum Profit Problem. [Adopted from Mujtaba and Macchietto, 1997]...

Note that the results of the maximum profit problem obtained using the techniques presented above will be close to those determined by rigorous optimisation method (using the techniques presented in Mujtaba and Macchietto, 1993, 1996) only if the polynomial approximations are very good as were the case for the example presented here. Mujtaba et al. (2004) presented Neural Network based approximations of these functions. 

Mujtaba (1999) used profit as the performance measure of BED processes. However, other measures such as productivity could also be considered (Tran and Mujtaba, 1997 Mujtaba et al., 1997). With profit as the performance measure, the MDO problem (maximum profit problem) can be formally stated as ... 

Mujtaba (1997) used the maximum distillate problem to compare the performances of the two types of distillation columns (CBD and continuous). With the amount of initial charge and the feed flow rate fixed in a continuous column, the operation time (pass time) also becomes fixed. The performance measure using maximum distillate problem allows fixing of the operation time. Other types of optimisation problems such as minimum time or maximum profit problems (presented in the previous chapters) are not suitable for the purpose of comparing the performances of... 

Maximum profit problem In this problem neither batch time nor final concentration is fixed but a profit function involving batch time and final... 

Maximum Profit Problem - where a profit function for a specified concentration of distUlate is maximized [53, 49, 54, 55]. 

An efficient optimization approach for reactive batch distillation using polynomial curve fitting techniques was presented by [55]. After finding the optimal solution of the maximum conversion problem, polynomial curve fitting techniques were applied over these solutions, resulting in a nonlinear algebraic maximum profit problem that can be efficiently solved by a standard NLP technique. Four parameters in the profit function, which are maximum conversion, optimum distillate, optimum reflux ratio, and total reboiler heat load, were then represented by polynomials in terms of batch time. This algebraic representation of the optimal solution can be used for online optimization of batch distillation. 

Optimization of a process is an activity whereby the best conditions are found for attainment of a maximum or minimum of some desired objective. In the broadest sense, an industrial process has maximum profit as its goal, but there are also problems with less-ambitious goals that do not involve money or the whole plant. 

Cost of recovery 2.2 + 1.8/J1-xa, /kmol of A The problem is to find the feed rate na0 and the fractional conversion xa for the maximum profit rate. 

Optimization pervades the fields of science, engineering, and business. In physics, many different optimal principles have been enunciated, describing natural phenomena in the fields of optics and classical mechanics. The field of statistics treats various principles termed maximum likelihood, minimum loss, and least squares, and business makes use of maximum profit, minimum cost, maximum use of resources, minimum effort, in its efforts to increase profits. A typical engineering problem can be posed as follows A process can be represented by some equations or perhaps solely by experimental data. You have a single performance criterion in mind such as minimum cost. The goal of optimization is to find the values of the variables in the process that yield the best value of the performance criterion. A trade-off usually exists between capital and operating costs. The described factors—process or model and the performance criterion—constitute the optimization problem. ... 

The objective function can assume different representation with regards to the system under study. A commonly used objective of an industrial process is to maximize profit or to minimize the overall costs. The former is adopted in this work. In this model, the whole refinery is considered to be one process, where the process uses a given petroleum crude to produce various products in order to achieve specific economic objectives. Thus, the objective of the optimization at hand is to achieve maximum profitability given the type of crude oil and the refinery facilities. No major hardware change in the current facilities is considered in this problem. The... 

The performance criteria of a batch distillation column can be measured in terms of maximum profit, maximum product or minimum time (Mujtaba, 1999). In distillation, whether batch, continuous or extractive, purity of the main products must be specified as it is driven by the customer demand and product prices. The amount of product and the operation time can be dictated by economics (maximum profit) or one of them can be fixed and the other is obtained (minimum time with fixed amount of product or maximum distillate with fixed operation time). The calculation of each of these will require formulation and solution of optimisation problems. A brief description of these optimisation problems is presented below. Further details will be provided in Chapter 5. 

Kerkhof and Vissers (1978) combined the minimum time and the maximum distillate problems into an economic profit function P to be maximised. 

Note that other types of dynamic optimisation problems such as minimum time, maximum profit, etc. could also be formulated and solved using the algorithms... 

Using the above profit function, the solution of problem P2 will automatically determine the optimum batch time (tf), conversion (C), reflux ratio (r) and the amount of product (Di). However, as the cost parameters (CDh CB0, etc.) can change from time to time, it will require a new solution of the dynamic optimisation problem P2 (as outlined in Mujtaba and Macchietto, 1993, 1996), to give the optimal amount of product, optimal batch time and optimal reflux ratio. And this is computationally expensive. To overcome this problem Mujtaba and Macchietto (1997) calculated the profit of the operation using the results of the maximum conversion problem (PI) which were obtained independent of the cost parameters. 

Case 1 of Table 9.3 is the base case. It shows the optimisation results using the cost parameters presented in Table 9.2. The maximum profit and optimal batch time obtained by optimisation shows very good agreement to those shown in Figure 9.8. The maximum profit shown in Figure 9.8 is between 3.99-4.13 ( /hr) with an optimum batch time between 12-14 hr. Each of the optimisation problems (i.e. solution of P2 with Equation 9.6) presented in Table 9.3 requires approximately 3- 4 iterations and about 3- 4 cpu sec using a SPARC-1 Workstation (Mujtaba and Macchietto, 1997). 

Problem 9.13 Optimum Cycle Time for Maximum Profit... 

Fig. 1 gives some geometric intuition about the mathematical structure of the problem and the way this structure can be used to find an optimal solution. The profit function is a plane and the highest point is the vertex A = 10, B = 20 with a profit of P = 110. The intersection of the profit function and planes of P = constant gives a line on the profit function plane as shown for P = 96. This diagram emphasizes the fact that the profit function is a plane, and the maximum profit will be at the highest point on the plane and located on the boundary at the intersection of constraint equations, a vertex. 

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