Profit function

Principles of mathematical modelling 2 Probability density function 112 Process control examples 505-524 Product inhibition 643, 649 Production rate in mass balance 27 Profit function 108 Proportional... 

The objective function for the literature example is the maximization of a profit function over a 10 h time horizon that takes revenue, freshwater and wastewater treatment costs as follows. 

The objective function can take two forms depending on the production information given. If the required production is known, then the objective is the minimisation of effluent. If this is not the case then the objective function takes the form of a profit function, where profit is dependent on the revenue from product, the cost of the raw material and the treatment costs of the effluent. 

The authors develop their model on the basis of these assumptions, inferring the demand and profit functions and the condition that the market is covered. Then they find the collective welfare. Under the conditions they set, a drop in the intervened price would bring about an increase in welfare, which... 

At least one objective function to be optimized (profit function, cost function, etc.). 

Set up the linear profit function and linear constraints to find the optimum product distribution, and apply the simplex technique to obtain numerical answers. 

The iterative program incorporating the quadratic interpolation search yielded the results in Table E12.4B. The optimum reflux ratio was 17.06 and the cost,/, was 3870/day. Table E12.4C shows the variation in/ for 10 percent change in R. The profit function changes 100/day or more. 

The expected profit determines the average profit across all price scenarios weighted with their scenario probability. The expected profit function can be defined as follows ... 

Optimisation may be used, for example, to minimise the cost of reactor operation or to maximise conversion. Having set up a mathematical model of a reactor system, it is only necessary to define a cost or profit function and then to minimise or maximise this by variation of the operational parameters, such as temperature, feed flow rate or coolant flow rate. The extremum can then be found either manually by trial and error or by the use of numerical optimisation algorithms. The first method is easily applied with MADONNA, or with any other simulation software, if only one operational parameter is allowed to vary at any one time. If two or more parameters are to be optimised this method becomes extremely cumbersome. To handle such problems, MADONNA has a built-in optimisation algorithm for the minimisation of a user-defined objective function. This can be activated by the OPTIMIZE command from the Parameter menu. In MADONNA the use of parametric plots for a single variable optimisation is easy and straight-forward. It often suffices to identify optimal conditions, as shown in Case A below. 

As many other industries, the fine chemical industry is characterized by strong pressures to decrease the time-to-market. New methods for the early screening of chemical reaction kinetics are needed (Heinzle and Hungerbiihler, 1997). Based on the data elaborated, the digital simulation of the chemical reactors is possible. The design of optimal feeding profiles to maximize predefined profit functions and the related assessment of critical reactor behavior is thus possible, as seen in the simulation examples RUN and SELCONT. 

The reactor design problem considered by Murase et al. (1970) is to maximize the following profit function ... 

The optimal control of a process can be defined as a control sequence in time, which when applied to the process over a specified control interval, will cause it to operate in some optimal manner. The criterion for optimality is defined in terms of an objective function and constraints and the process is characterised by a dynamic model. The optimality criterion in batch distillation may have a number of forms, maximising a profit function, maximising the amount of product, minimising the batch time, etc. subject to any constraints on the system. The most common constraints in batch distillation are on the amount and on the purity of the product at the end of the process or at some intermediate point in time. The most common control variable of the process is the reflux ratio for a conventional column and reboil ratio for an inverted column and both for an MVC column. 

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

For single separation duty, Diwekar et al. (1989) considered the multiperiod optimisation problem and for each individual mixture selected the column size (number of plates) and the optimal amounts of each fraction by maximising a profit function, with a predefined conventional reflux policy. For multicomponent mixtures, both single and multiple product options were considered. The authors used a simple model with the assumptions of equimolal overflow, constant relative volatility and negligible column holdup, then applied an extended shortcut method commonly used for continuous distillation and based on the assumption that the batch distillation column can be considered as a continuous column with changing feed (see Type II model in Chapter 4). In other words, the bottom product of one time step forms the feed of the next time step. The pseudo-continuous distillation model thus obtained was then solved using a modified Fenske-Underwood-Gilliland method (see Type II model in Chapter 4) with no plate-to-plate calculations. The... 

For single separation duty, Mujtaba and Macchietto (1993) proposed a method, based on extensions of the techniques of Mujtaba (1989) and Mujtaba and Macchietto (1988, 1989, 1991, 1992), to determine the optimal multiperiod operation policies for binary and general multicomponent batch distillation of a given feed mixture, with several main-cuts and off-cuts. A two level dynamic optimisation formulation was presented so as to maximise a general profit function for the multiperiod operation, subject to general constraints. The solution of this problem determines the optimal amount of each main and off cut, the optimal duration of each distillation task and the optimal reflux ratio profiles during each production period. The outer level optimisation maximises the profit function by... 

Note, the profit function can be as general as desired (Logsdon et al., 1990), to reflect both annualised investment (e.g. column, reboiler, condenser, etc.) and operating cost (e.g. steam). 

Therefore only two time periods are involved in the optimisation problem formulation, one for each mixture. Instantaneous switching occurs between batches and therefore set up time between the batches is ignored. The profit function does not include the allocation of time to each separation. For each mixture (m) individual profits (Pm) were maximised to maximise the overall profit. 

This example is taken from Mujtaba and Macchietto (1996). Here, the profit function (Equation 7.27) used by Logsdon et al. (1990) is considered for the multiple separation duties presented in section 7.3.4.2. Using the input data presented in Table 7.3 with (0, = 02 = 0 and tsul = tm2 - 0 hr) the optimal design and operation policies are obtained and the results are presented in Table 7.7. The profit per batch for each separation is calculated by multiplying the profit per hr and the batch time. 

It is important to note that the use of Logsdon et al. s objective function will not change the number of plates and operating policies if the mixtures were to share the column by a certain fraction of time (the profit function of Logsdon et al. does not include the allocation time). However, for different allocation time to each mixture, the total number of batches for each separation, individual and total yearly profit, etc. can be evaluated by using the results presented in Table 7.7. These are summarised in Table 7.8. Comparison of the results in Table 7.4 and 7.8 clearly... 

Mujtaba and Macchietto (1997) defined a typical profit function P as ... 

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. 

The profit shown in Table 11.8 is calculated using the profit function defined... 

Refinery profit as measured by the LP objective function increased as the incremental crude was charged to the refinery. At a point equal to about 50% of the available incremental crude, however, the profit function began to decline. The profit at maximum incremental crude rate actually ended up at a lower value than the initial LP case value using this parametric approach. 

FIGURE I Profit functions for EPO (A) power-law model and (B) second-order polynomial. (Data taken from Furst.11)... 

Decide on the units to be used for expressing the profit function for each individual stage and the overall process. For this case, the problem statement makes it clear that an appropriate unit for this purpose is the profit over a... 

To determine the optimum value of kgX Z, we take the derivative of the Profit function (or, if one wishes the raw material cost function), set it equal to zero, and solve for kgX Z... 

Unlike the past work, this work focuses on optimal design and operation of multivessel batch distillation column with fixed product demand and strict product specifications. Both the vapour load and number of stages in each column section are optimised to maximise a profit function. For a ternary mixture, the performance of the multivessel column is also evaluated against that of a conventional batch distillation column. Although the profitability and the annual capitalised cost (investment) of the multivessel column is within 2-3% compared to those of conventional column, the operating cost (an indirect measure of the energy cost and environmental impact) is more than 30% lower for multivessel column. Thus, for a given separation task, multivessel column is more environment friendly. 

Low and Sorenson (2003) presented the optimal design and operation of MultiVBD column. A profit function based on revenue, capital cost and operating cost was maximized while optimising the number of stages in different column sections, reflux ratio, etc. They compared the performance of MultiVBD with that of conventional batch distillation column for a number of different mixtures and claimed that MultiVBD operation is more profitable. However, for all cases considered in their work, the products specifications and amounts were not matched exactly and therefore the... 

The optimization of any industrial process aims the profit maximization. Thus, the profit function is a natural choice as an objective function. The profit function, as outlined by Xiong and Jutan [4] and Sequeira et al. [5], can be calculated based on the selling price of the products and on the costs of raw materials, operation and energy. Then, in this work, the objective function, adapted to the multiphase reactor, is as follows ... 

Profit function prices, economic, environmental and sustainable costs... 

Maximize the profit function to find the optimum process configuration with the system. 

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