Profitability functional products

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 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. 

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 objective function is to maximize profits, namely, products sold minus raw material costs. No capital or investment cost are involved in this example. 

Because the commercial development activity, particularly in the field of plastics, is so frequently misunderstood, or little understood, this symposium was undertaken with an aim toward clarifying the activities involved in developing a new plastic to a profitable commercial product. This symposium was presented on a how-to-do-it basis and is composed of a series of papers by experts in each of the fields and functions covered. Each author is eminently qualified because of his years of successful experience in this particular area of expertise. Prices referred to were those in effect in early 1969. 

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. 

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... 

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. 

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 ... 

In terms of the present methanol study, the problem seeks an optimum allocation policy for corn stover (i 1), furfural residue (i = 2), and wood residue (i = 3) to produce three commodity products of electricity (j = 1), transportation (j = 2). and chemicals (j = 3) in the most profitable way. In pure mathematical terms, we are to determine fjj which maximizes the profit function. Equation 1, when the feedstock availability and demand are constrained by Equations 2 and 3, respectively. The data base to use with Equations 1 through 4 is tabulated in Table I for Fj. Table II for Dj and Sj, Table III for ny, and Table IV for My. 

These decisions have to be optimal with respect to a certain objective or profit function. Such a profit function contains the profit resulting from the conversion of A (e.g., SO2) into the product P (e.g., SO3X but also the costs (catalyst, construction, control. ..). If the costs were not taken into account it would follow from the computations that the conversions should proceed to the equilibrium values, and this would require an infinite amount of catalyst. Let a represent the profit resulting from the conversion of 1 kmol of A into P. (e.g., U.S. /kmol). Per stage the value of the reaction mixture increases by an amount (in per hr) ... 

A piecewise linear function arises when the per-unit contribution (cost) depends on the level of sales (production). For example, consider a product whose profit contribution is 10/unit for the first 40 units, 8/unit for the next 60 units, and 5/unit for the rest. The nonlinearity of the profit function is apparent if a graph is plotted between total profit and quantity sold. This is illustrated by Figure 1, which is called a piecewise linear function since it is linear in the region (0, 40), (40, 100), and (100,oo). Partitioning the quantity sold into three activities means that the profit function could be expressed as a linear function as follows ... 

The ranges on the objective function coefficients given in Table 3 exhibit the sensitivity of the optimal solution with respect to changes in the unit profits of the three products. It shows that the optimal solution wiU not be affected as long as the unit profit of product 1 stays between 6 rmd 15. Of course, the maximum profit will be affected by the change. For example, if the unit profit on product 1 increases from 10 to 12, the optimal solution will be the same, but the maximum profit will increase to 733.33 + (12 - 10) (33.33) = 799.99. 

The other dimension in Figure 9.3 is value. Value, in this application, is the product s importance either to the customer or to the company. High-value products sell for high prices and generate profits for the company. As described in Section 5.1.3, innovative products, by definition, are high value (quadrants I and II) while functional products are often low value (quadrants III and IV). Innovative products need responsive supply chains while functional products need efficient ones. 

Section 5.1.3 described the differences between functional and innovative products. Applying this model, supply chains for functional products must be efficient. So cost is a dominant feature and determinant of supply chain design success. On the other hand, supply chains for innovative products must be responsive. The emphasis is on making high-profit products readily available to the market and minimizing lost sales to supply-demand mismatches. 

Acceptance Wage s the shadow value of the iso-utility function for workers with risk aversion given by a particular value of a and tc = 0. Offer Wage = the iso-profit function value for firms with safety productivity given by a particular value of p and 7i = 0. 

As with any system of differential equations boundary conditions are necessary to determine a unique solution. The assumptions made concerning the expected profit function, G n, n, k wi), imply that the productivity parameter, p, increases both profit and the optimal level of safety. Thus, the zero profit constraint endogenously determines the minimum productivity of safety equipment in the production of goods... 

Having presented in Appendix 3A the technical details of how the labor market establishes hedonic equilibrium we now fill in the details of the structural equations underlying the matching of workers to jobs by injury risk. We fill out the structure of our numerical simulations by describing the quantitative properties of workers utility functions including the distribution of workers by risk preferences, firms profit functions including the distribution of employers by their costs of injury reduction, and the economic properties of product and input markets. 

Here we provide more details of the structure of our simulation model. The discussion to follow describes our numerical specification of the three key components of the model, which are workers utility functions and heterogeneity in attitudes towards risk, firms profit functions and heterogeneity in ability to jointly produce output and workplace safety, and the characteristics of how people and firms interact in the labor and product markets. 

Safety programs are jointly productive inputs additional safety equipment increases output and decreases deaths due to industrial injuries. The parameter p determines the magnitude of the output effect. Firm heterogeneity is captured by assuming p is uniformly distributed from 0 to 1. We took all other expected profit function parameters to be identical across firms. It is important to recognize that... 

Bhatagar et al. [44] raised a question about coordination of multi-plants production in a vertically integrated firm, and had a thorough review of conventional coordination problems and multi-plants coordination problems. Chien [45] smdied a similar problem in [43] considering stochastic demand conditions. Based on the stable and independent weekly demand and the known probability distribution, they derived the average profit function of the unit product production and transportation and then got the optimal production-transport strategy by means of the analytic method. The numerical example verified the validity of the model based on Monte Carlo (Monte Carlo) simulation and sensitivity analysis. 

Objective function (4.15) shows that when the demand and price of external finished products in the supply chain are random variables, the objective of the supply chain is to maximize the expected revenue. The profit function consists of three parts of suppliers profit, manufacturer profit and customers profit fo, which will be explained in details below. 

The values of Bi are relative to the profit per product (in 1). The evaluation function has to take into account the constraint Dacompletion time of all products Da must be achieved at the end of time horizon H. For this purpose, the following expression has been considered ... 

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