Economic Objective Function In Process Flow Sheet Optimization – Do We Know Everything about It?

Introduction

Design and synthesis of process flow sheets by using mathematical programming approach are considered difficult, especially if large-scale industrial processes are concerned or additional challenges are included, e.g. uncertainty for flexible design. Over the last three decades, the process engineering community put tremendous effort into developing a) modeling approaches to deriving appropriate mathematical formulations of such processes, and b) solution strategies for solving mathematical models efficiently to optimum solutions. Less attention has been paid to the type and correctness of the economic criteria used in the objective functions of flow sheet optimization models.

The annual profit, the total annual cost, the payback time, the net present value (NPV), and the internal rate of return (IRR) are the economic criteria that are more often applied to flow sheet design and synthesis. Many authors have observed that these measures generate different optimal flow sheet solutions when used as decision criteria, e.g. Faria and Bagajewicz (2009), Bagajewicz (2008). In our previous work (Novak Pintaric and Kravanja, 2006) it was established that economic criteria can be roughly classified into three groups regarding the optimal results generated: quantitative criteria which are expressed in monetary units (e.g. profit, cost), qualitative criteria which are expressed in non-monetary units (e.g. IRR, payback time), and compromise criteria, e.g. the NPV, which are the compromise between qualitative and quantitative measures.

In recent year, a further research has been carried out which revealed some fundamentally new understandings about the correctness of various economic criteria, and the consequences of their use in optimization-based flow sheet synthesis and design (Kasaš et al. 2011a, 2011b). To the best of our knowledge, some of the following questions have been cleared up for the first time: a) which economic criteria are the correct ones, b) what is the reason for the differences between those optimal solutions obtained by different criteria, and c) what are the consequences of using inappropriate criteria in the objective functions of flow sheet optimization models.

The correctness of economic criteria in process flow sheet optimization 

In the financial theory, projects are classified into independent and mutually exclusive ones. Both groups of projects have specific approaches of how to select the best alternative(s). It is important to realize that flow sheet design and synthesis by means of mathematical programming belong to the selection between mutually exclusive alternatives. The capital investment is very often considered unconstrained, as it is explicitly assumed that funds are available for realization of selected optimal design. According to the finance and accounting theory, the net present value is the correct criterion for such type of investment projects, as it uses all the cash flows of the project and discounts the cash flows properly. Other criteria are not totally correct, e.g. mutually exclusive alternatives cannot be compared by their individual IRRs, as they do not take into account the level of investment, and mistakenly favor projects with small investment. A similar effect was observed in the case of the payback time criterion. Incremental analysis of IRR is needed, where the IRR of each increment of investment must satisfy the minimum acceptable rate of return (MARR) requirement. Criteria like total cost and profit are the accounting measures, which are not based on the cash flows, and are thus inappropriate for investment decision making.

The above mentioned criteria would, in general, generate different optimal results when used in flow sheet optimization problems as the objective functions. Moreover, the consequences of using inappropriate criteria could be far more serious than it might seem at first glance.

The source of the differences between optimal flow sheet designs

The most obvious reason for differences between optimal flow sheet designs obtained by using various economic criteria is that the stationary conditions of these criteria are different. For example, the maximum NPV at unconstrained investment is obtained at an investment level where the incremental NPV equals zero, and incremental IRR equals the MARR. The maximum profit before taxes and the maximum IRR are obtained at those investment levels where the incremental profit and IRR, respectively, become equal to 0. It could be shown, by analyzing stationary conditions regarding the relation between the cash flow and investment, that qualitative criteria, e.g. IRR, generate optimal solutions at the lowest investment levels, while the quantitative criteria, e.g. profit and cost, at the highest investment levels. The investment levels of those solutions obtained by compromise criteria like the NPV are in-between, as the NPV establishes a balance between the profitability of investment and long-term steady generation of cash flow.

In spite of well-defined differences in stationary conditions, some process flow sheet models produce negligible differences when optimized with different objectives, while other models produce substantially-different optimal designs. A source of this apparent contradiction is in the accuracy and preciseness of flow sheet model regarding the trade-offs between the investment and cash flow generated. These trade-offs are often insufficient if simple and aggregated models are used, e.g. stoichiometric model for the reactor or simple component splitter for the separation. More precise trade-offs are present in more detailed modeling where increased investment level results in increased benefit, e.g. higher conversion in the kinetic reactor, higher purity of main product in more precise separator model etc. The more precise correlations between the investment and resulting gains are included in the model, the larger differences in optimal results could be expected.

It was shown (Kasaš et al. 2011a) that the steepness of the cash flow derivative curve vs. investment is the main responsible for the magnitude of the differences between optimal results obtained by different economic criteria. The more gentle derivative function of cash flow vs. investment, the larger differences would be obtained. Less precise models generate unimodal cash flow functions with maximum, whose derivative curves are very steep. Optimal solutions obtained by different criteria are thus similar or equal. More precise models generate monotonically increasing cash flow functions and flat derivative curves, optimal solutions are thus further apart.

The consequences of using different economic criteria

The consequences of using different economic criteria in process flow sheet optimization are numerous, and can be classified into economic, operational and environmental. Economic consequences follow straightforwardly from the facts given in previous subsection. The investment levels and cash flows of optimal solutions increase from the qualitative criteria (IRR, payback time) over the compromise (NPV) to the quantitative criteria (profit, cost).

The operational consequences can be evaluated through the overall operational efficiency of optimal solutions, which could be expressed as a financial ratio between the net income and sales. It was proved that operational efficiencies grow from the IRR, over the NPV, to the profit solution (Kasaš et al. 2011b). IRR criterion leads to inefficient utilization of reactant and utilities, which reflects in lower conversion in the reactor, lower purity of the product, lower level of heat recovery etc. Maximization of profit produces the most efficient solutions with the lowest operating costs, while the operational efficiencies of optimal NPV solutions are in-between.

Optimal solutions are different also regarding the environmental performance. The latter can be evaluated using various indicators, e.g. global warming, acid rain potential etc. Mixed integer nonlinear programming (MINLP) syntheses of heat integrated processes (methanol, dimethyl ether, and HDA) showed that profit criterion generates environmentally most conscious solutions, while the environmental impacts of the IRR solutions are the highest. Optimal NPV designs are the environmental compromises between the other two solutions.

Conclusions

This contribution summarizes the outcomes of the research on the economic criteria used in process flow sheet optimization, and points out some important findings that have been often overlooked by now. The NPV is the correct criterion for optimization-based flow sheet design and synthesis, while other measures like profit and internal rate of return are not applicable decision criteria. The insufficient trade-offs in too simplified and aggregated models could prevent from generating proper optimal solutions even with the correct optimization criterion. By using the NPV criterion in precisely defined process flow sheet models, a well-balanced compromise between the economic performance, operational efficiency, and environmental impacts can be achieved during a single-criterion optimization. Such solutions represent correct compromise designs in the financial and environmental sense, as well as in the sense of efficient revenue generation and operating cost control.

References

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Kasaš M., Kravanja Z., Novak Pintariè Z. Suitable modeling for process flow sheet optimization using the correct economic criterion. Ind. Eng. Chem. Res. 2011a, 50, 3356-3370.

Kasaš M., Kravanja Z., Novak Pintariè Z. Achieving profitably, operationally and environmentally compromise flow-sheet designs by a single-criterion optimization. Submitted to AIChE J. 2011b.

Novak Pintariè Z., Kravanja Z. Selection of the economic objective function for the optimization of process flow sheets. Ind. Eng. Chem. Res. 2006, 45, 4222-4232.