Plant-Wide Waste Management. 3. Long-Term Operation and

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Ind. Eng. Chem. Res. 2003, 42, 4772-4788

Plant-Wide Waste Management. 3. Long-Term Operation and Investment Planning under Uncertainty Aninda Chakraborty,† Richard D. Colberg,‡ and Andreas A. Linninger*,† Laboratory for Product and Process Design, Department of Chemical Engineering, University of Illinois, Chicago, Chicago, Illinois 60607, and Chemical Intermediates & Catalysis Research Laboratory, Eastman Chemical Company, Kingsport, Tennessee 37662

Pollution prevention and waste reduction efforts at pharmaceutical sites with ever-changing production campaigns require a dynamic model of the corporate business and manufacturing activities. Flexibility in business strategies for addressing evolving market demands or changes in environmental legislation add uncertainty to this already challenging design problem. In this third part of the triple series on plant-wide waste management, we extend the combinatorial process synthesis methodology to the computer-aided synthesis of long-term site-management strategies under uncertainty. The proposed framework formulates optimal management of operations and facilities as a multiperiod optimization problem with a planning horizon of typically 5 years. Our approach synthesizes a network of process operations to recover useful raw materials and solvents, treat unavoidable effluent streams to compliance, and suggest optimal capacity and timing of investment decisions for new facilities or process technology. In addition, the expected environmental discharges and emissions are projected for the entire planning horizon. The mathematical modeling and solution approach allows decision makers to assess the monetary, infrastructural, and ecological impact of new environmental legislation or changing production requirements on their businesses in advance. The methodology is validated with an industrial-type case study. 1. Background and Overview A novel combinatorial process synthesis methodology was introduced in the first part of this series.1 This methodology was aimed at constructing optimal recovery and treatment policies for entire manufacturing sites using a two-step procedure. In the first stage, superstructure synthesis, a linear planning algorithm generated a network of feasible recovery and treatment options for all unavoidable byproduct streams at the manufacturing site. The recovery tasks recycle valuable raw materials or solvents trapped in the effluent streams via physical separations such as distillation, decantation, extraction, or stripping. Destructive treatment renders a waste stream less harmful by chemically destroying hazardous or toxic compounds. Treatment options include incineration, wet-air oxidation, and biological or chemical treatment via neutralization or scrubbing. Execution of each possible step led to an acyclic directed graph storing all feasible treatment alternatives for each effluent stream, the amount and composition of the residuals, as well as the operating cost for each treatment step. Permutations of the treatment alternatives for each effluent constitute plantwide treatment policies. Each policy is composed of an ordered sequence of reaction or separation tasks transforming each unavoidable waste into compliant residuals. The union of distinct policies constitute the superstructure of all possible recovery and treatment options. In the second stage, superstructure optimization, a largescale mathematical program searched for the best * To whom correspondence should be addressed. Tel.: (312) 996-2581. Fax: (312) 996-0808. E-mail: [email protected]. † University of Illinois, Chicago. ‡ Eastman Chemical Company.

operating policy embedded within the superstructure. We developed plant-wide recovery and treatment policies with optimal tradeoff between conflicting objectives such as cost and environmental impact. In the second part of the series,2 we extended our planning methodology for optimizing operating procedures in the presence of uncertainty in the effluent streams. We demonstrated a mathematical programming framework for designing plant-wide policies with the desired degree of process flexibility. Multi-objective programming techniques balanced economics, ecology, and robustness against uncertainty.3,4 In the two earlier approaches, the waste management problem was solved for a fixed plant inventory at steady state. A dynamic view for designing optimal long-term waste management strategies requires models for changes in future production plans as well as investment decisions for augmenting the plant infrastructure in time. Optimal planning of chemical plant operations and investment schedules for the future can be formulated as a multiperiod optimization problem. Multiperiod optimization can be classified into three major categories:5 (i) multiperiod design, (iii) multiperiod planning, and (ii) multiperiod design with capacity planning. The goal in multiperiod design6-8 is to decide on the values for design and state variables for a plant with minimum cost but feasible operation in all time periods. Multiperiod planning considers decisions involving startup/shutdown of operations for a fixed plant inventory. Design with capacity expansions for manufacturing processes under uncertainty is a challenging problem that is receiving ample attention in the open literature.9-13 Clay and Grossmann14 proposed a sensitivity-based successive disaggregation algorithm without considering

10.1021/ie0210614 CCC: $25.00 © 2003 American Chemical Society Published on Web 09/04/2003

Ind. Eng. Chem. Res., Vol. 42, No. 20, 2003 4773

Figure 1. Information flow diagram of the proposed multiperiod planning methodology.

capacity expansions. Other stochastic planning models deployed Gaussian quadrature to compute expected cost.15 Monte Carlo sampling (MCS) techniques are reported to reduce the computational effort for uncertain variables.16 Future demands and prices of chemicals over a long-range horizon were modeled by time-varying forecasts.11 Sahinidis and co-workers also proposed several programming strategies to reduce the computational expense for solving such problems, e.g., branch and bound, cutting planes, and Bender’s decomposition algorithm. They extended their model to include investment decisions for the processing network with dedicated and flexible plants.17 Paules and Floudas18 designed a flexible sequence of distillation columns from a multiperiod superstructure, which could handle prespecified changes in feed streams for a finite number of time periods. In multiperiod planning, open variables include structural decisions for operational tasks as well as new process equipment. Despite the attention devoted to design, less work is dedicated to multiperiod planning with capacity expansion. Planning of operations and capacity augmentation for multipurpose batch manufacturing sites has not yet been addressed to the best of our knowledge. Accordingly, the last of this triple series deals with long-term planning of recovery and treatment operations together with process capacity expansion and equipment investments for optimal waste management at multipurpose pharmaceutical manufacturing sites. Section 2 will introduce the basic elements of the longterm investment planning methodology. It will introduce techniques to derive future production plans and their associated discharges from market and business forecasts. The stream table of effluents called the waste scenario will be used to generate a superstructure of treatment options. Section 3 proposes a multiperiod mixed-integer programming framework for finding op-

timal site-wide waste management strategies together with corresponding investment decisions necessary to sustain the projected production plans. We will demonstrate mathematical formulations for modeling the current plant inventory as well as equipment investments. The regulatory forecasts will provide information on expected or possible changes in environmental legislation. Section 4 validates the long-term planning methodology via an industrial-type case study. In section 5, the methodology will be extended to include uncertainty in the waste forecasts. A second case study will elucidate the impact of uncertainty in these forecasts. The paper will close with conclusions and significance. A numerical example of the multiperiod optimization framework will be given in the appendix. 2. Methodology Pharmaceutical companies are constantly engaged in research and development of new drugs. The time to market for a new medication typically consumes 6-12 years. The first company to achieve approval by the Food and Drug Administration for the new drug and its batch manufacturing recipe can typically expect to gain 60% of the market share for the application. Long development times and high risk are rewarded by a sales income on the order of $1M/day for a lucrative drug on the market. To address the time-varying nature of this business, this section will expand the combinatorial process synthesis methodology to create optimal long-term waste management strategies for a planning horizon of typically 5 years. Future trends in the desired production capacities will be derived from the market and business forecasts. The information flow of the proposed planning tool is depicted in Figure 1. The products and their desired production level give rise to expected waste loads and their compositions, the so-called waste fore-

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cast. With the waste forecast as input, the superstructure generation step synthesizes feasible recovery and treatment options for each time period of the planning horizon. Superstructure optimization finds optimal waste management strategies minimizing total annualized cost by fixing the operating policies for each time period. The optimization via mixed-integer programming techniques sets decision variables choosing the best sequence of recovery and treatment tasks and determines continuous variables for the respective operation conditions. In addition, investment decisions for new reactors or separators may be needed to sustain the corporate production objectives in the years to come. We will first propose models for anticipating production requirements for the site via different forecasting techniques. Then we will proceed to discuss the mathematical framework for optimal pharmaceutical site management with capacity expansion. 2.1. Superstructure Generation for Future Production Plans. The market forecast examines the company’s current product portfolio as well as market analysis of demands for existing pharmaceutical products. Corporate decision makers formulate a business forecast in response to the market demands and expected new products coming out of the corporate’s research and development pipeline. The information contained in the business forecast translates into a corporate production plan for the coming time period. Because intentional business plans cannot be predicted by mathematical means, qualitative judgmental techniques are deployed.19,20 These techniques lead to different scenarios under optimistic, conservative, or pessimistic assumptions quantified by a panel of experts, e.g., the Delphi method,21,22 ABC approach,23 etc. In this paper, the business forecast is treated as known input to the combinatorial process synthesis methodology. Waste Forecast. Future business activities alter the discharges of unavoidable effluents. For any given time period, waste forecasts specify loads and probability for all expected byproducts and effluents from all production campaigns at the site. All state parameters of the waste stream may vary, e.g., load per batch, composition, pressure, and temperature. The initial stream table collecting the information on all unavoidable byproduct streams is represented by the set W(0) ) {w1(0), w2(0), ..., wN(0)}. The initial waste stream table can be inferred from up-to-date production data, batch records, or through the simulation of batch manufacturing campaigns via specialized software, e.g., Batch Design Kit.24 Future discharges of effluent streams are quantified by the waste forecast, W(t) ) {w1(t), w2(t), ..., wN(t)}. Statistical techniques such as the Time Series Models25,26 predict future waste loads over the complete planning horizon based on historical data. For more realistic projections, the simple time series pattern can be combined with random fluctuations and/or seasonal trends superimposed with probable variations. Waste Categories. At large production sites with dozens of products from numerous campaigns in the course of a year, thousands of waste streams may arise. In that situation, it is often not practical to account for every particular waste stream. A significant simplification can be achieved by lumping waste streams with similar properties. Table 127 proposes a classification of seven waste categories. Waste streams with comparable properties are likely to be treated in a similar

Table 1. Waste Categorization for Multiperiod Planning type wastewater burnable liquids nonburnable liquids burnable solids nonburnable solids burnable vapors nonburnable vapors

properties low salt, high salt, low solids, high solids, low volatility content, high volatility content hazardous, nonhazardous, chlorinated, nonchlorinated, wet, dry hazardous, nonhazardous, high/low Btu content hazardous, nonhazardous, heat of combustion hazardous, nonhazardous, trash, toxicity hazardous, nonhazardous, HAPs, VOC hazardous, nonhazardous

fashion; hence, waste categorization may often render mathematically tractable problem sizes without oversimplification. 2.2. Synthesis of Treatment Options: Superstructure Generation. The waste scenario provides the state information about each waste stream for each year. Even though the particular waste loads vary from year to year, we assume the compositional changes in each waste category to be small enough that they do not affect the selection of feasible recycle and treatment options (see Figure 1). Consequently, the superstructure generated for the first year will also accommodate the set of possible recycle and treatment steps for future time periods. Only the computation of residuals and treatment cost must be adjusted for varying waste loads, a task greatly simplified by performing dynamic linearization as described in part 2 of this series.2 Accordingly, the actual flow rates sent through the network of alternative recycle and treatment options are updated. As a consequence, the superstructure can be considered time invariant for the entire planning period. We will show next how to arrive at optimal treatment policies and equipment investment decisions. This task is accomplished by solving a multiperiod optimization problem as described in the subsequent section. 3. Mathematical Framework for Multiperiod Decision Making The multiperiod superstructure described above embeds implicitly all waste management strategies for the planning horizon of 5 years (n ) 5). Typically, we seek the strategy that minimizes the total net present cost (NPC) by selecting the best operating policy for each time period. In addition, capital investments may augment the available plant infrastructure of reactors and separators. Investments can only be made during the planning horizon.28 After the expiration of the planning horizon, all inputs are fixed at their last year value and no more capital investments are allowed. An extended economic horizon of 20 years is necessary to correctly assess the long-term impact of investment decisions during the shorter planning horizon. The roles of planning and economic horizon are similar to the control and prediction horizon in the model predictive control algorithm.29 The next subsection introduces a mathematical model for the proposed multiperiod program including (i) cost models accounting for annualized operating, investment, and maintenance cost, (ii) the optimal assignment of operational tasks into the site’s inventory of processing units, and (iii) an approach to make the waste management strategies compliant with current and future regulations. 3.1. Process Economics. In the multiperiod mixedinteger linear program (MILP) of Problem A, 1-14, the

Ind. Eng. Chem. Res., Vol. 42, No. 20, 2003 4775 Table 2. Typical Off-Site Cost and Disposal Methods for Different Waste Blends43 type of waste

specific cost ($/lb)

bulk organic liquids sludges (with organics) sludges (with inorganics)

0.35 1.15 0.43

solids (with organics) solids (with trace organics)

0.85 0.13

disposal method incineration incineration stabilization + secure landfill incineration secure landfill

Table 3. Investment Portfolio for a Solvent Recovery Plant: Installed Cost ($) of Distillation Columns (C1-C9: Column Types)a

vector, i.e., dmax T (0). The inventory of processing equipment types available at a particular site is stored in the plant model. The equipment purchases augment the level of available recovery and treatment capacity at the plant (T); cf. dmax T (t) in eq 7. The capacity expansion [∆dT(t)] equals the size of each module (Se) times the number of new units purchased [∆ne(t)] as given in eq 8.

Problem A: Min

x(t),ne(t),nT,e(t),qT(t),pT(t)

height diameter (in.)

22 ft

35 ft

82 ft

12 18 24 36

329 840

368 510 440 280 516 800

493 740 642 080 801 120

a

NPC ) NPCR + NPCβ + NPCγ (1)

N

200 ft

N

(1 + r)-tc(t)‚x(t) + ∑(1 + r)-t ∑ µTpT(t) ∑ t)0 t)0 ∀T∈I

NPCR )

T

(2) 1 432 230 2 182 878 correlation36

dT(t)‚x(t) ) dmax T (t) + qT(t)

Purchase cost was estimated using the Gutherie with M&S Index 2002. Installed cost ) purchase cost × Lang factor.

objective minimizes the NPC accounting for annualized capital investment, operating, and maintenance cost for the entire economic horizon by selecting operating procedures and proposing an optimal capital investment schedule. All cost considerations take into account the time value of money based on a NPC formula with interest rate r. In our case studies, we have used an interest rate of 10%.30 For simplicity, we incorporated the concept of time value of money without considering complexities such as inflation, income tax, or depreciation cost of capital. More sophisticated models based on total annualized performances or annuities are discussed elsewhere.31,32 Net Present Operating Cost. The net present operating cost, NPCR, in eq 2 consists of two contributions: cost for individual recovery and treatment steps and penalties for sending waste to off-site facilities. The operating cost vector c(t), whose values were determined in the superstructure generation phase by dynamic linearization, is a linear function of the waste loads.2 Hence, the treatment cost for varying effluent discharges in future planning periods can easily be computed. It contains operating cost information corresponding to all treatment paths marked by the binary decision variables x(t) embedded within the superstructure. The variable vector µT accounts for specific penalties for exceeding available capacities of the treatment plant (T) measured by the amount of excess material streams (pT). As an example, consider penalties accounting for additional transportation cost for off-site handling as given in Table 2. The level of excess discharge is computed with the help of slack variables (qT) in equations (3) and inequalities (4). The vector dT(t) represents the total waste load directed to the treatment plant, T. The waste load treated by process T must remain below its available capacity (dmax T ). Net Present Capital Cost. The net present capital cost of equipment investments, NPCβ (eq 5), computes the installed cost of newly purchased processing equipment. The total inventory of a particular equipment type (e) is modeled using integer variables ne(t) in eq 6. The cost of each module is Ce, while the integer variable ∆ne(t) denotes the number of new units of type e purchased in period, t. The initial capacity of each treatment plant is expressed by the available capacity

pT(t) g qT(t); pT(t) g 0

∀ T ∈ IT

(3)

∀ T ∈ IT

(4)

n

∑(1 + r)-t∀T∈I ∑ (∀e∈I ∑ Ce∆ne) t)0

NPCβ )

T

(5)

e

∆ne(t) ) ne(t) - ne(t-1); ∆ne(t) g 0 ∀ e ∈ Ie ∀ T ∈ IT

max dmax T (t) ) dT (t-1) + ∆dT(t)

∆dT(t) )

∑ ∆ne(t) Se

∀ e ∈ Ie; ∀ T ∈ IT

∀e∈Ie

(6) (7) (8)

N

(1 + r)-t ∑ ωT ∑ CenT,e(t) ∑ t)0 ∀T∈I ∀e∈I

NPCγ )

T

nT,e(t) e ne(t) ∑ ∀T dT(t)‚x(t) e nT,e(t) ) 0

(9)

e

∀ e ∈ Ie; ∀ T ∈ IT

∑ nT,e(t) Se

∀e∈Ie

∀ T ∈ IT

(10) (11)

∀ T: {is_feasible(e) ) false} (12)

E(t)‚x(t) e emax(t)

(13)

∑xj ) xi

(14)

∀ xj ∈ children(xi)

Economy of Scale for Equipment Purchases. For some equipment types, only discrete module sizes (Se) are offered by engineering companies. In general, it is also cheaper to buy one larger module than several small-sized units for the same total capacity. The economy of scale for equipment sizes typically follows a 6/10th rule.33 The investment models of Tables 3 and 4 quantify the influence of modularity and economy of scale effects.34 Net Present Maintenance Cost. The net present maintenance cost, NPCγ, in eq 9 factors maintenance charges for each processing unit. Dedicated units serving the specific needs of a decentralized process have higher maintenance costs than centralized facilities. The maintenance cost of dedicated equipment (e) are approximated by a constant fraction (ωT) of their purchase cost Ce(T) attributing proportionally higher cost to more

4776 Ind. Eng. Chem. Res., Vol. 42, No. 20, 2003 Table 4. Investment Portfolio for the Incineration Planta equipment

capacity

hazardous waste incinerator

30 MBtu/h (∼7500 lb/h)

installed cost ($) 3 195 780

50 MBtu/h (∼12 500 lb/h) 4 933 420 80 MBtu/h (∼20 000 lb/h) 7 356 140 120 MBtu/h (∼30 000 lb/h) 10 383 110 environmental control: 10 000 ft3/min 111 200 wet scrubber 20 000 ft3/min 222 900 30 000 ft3/min 335 000 a Purchase cost of incinerators estimated from the Gutherie correlation with M&S Index 2002. Installed cost of incinerators ) purchase cost × 5. Installed cost of scrubbers estimated using MATCHE Software38 (cost based on year 2000).

expensive equipment. The computation of maintenance cost is linked to the problem of assigning particular treatment/separation tasks to decentralized processing units; this constitutes a facility allocation task described next. 3.2. Facility Allocation for Recovery and Treatment Tasks. Optimal operation policies map all recycle and treatment steps into particular process equipment. The task allocation and capacity expansion model answers the following two questions: (i) Which equipment units out of the available plant inventory should be used to perform the desired treatment or separation task in a given time period? (ii) What matches between new processing units and tasks are optimal for future planning periods? For this end, additional integer variables nΤ,e(t) are introduced, each one denoting the number of units of particular type (e) assigned to a task (Τ) in any given time period (t). A typical example is the assignment of one or more distillation column(s) of a particular design (i.e., diameter and height) to perform a multistep separation task. The total columns assigned for distillation must not exceed their available number in the same period [i.e., ne(t) of inequality (10)]. Moreover, the cumulative waste loads sent to the selected units cannot surpass their maximum capacity as implied by inequalities (11). In the case of distillative separation, the total vapor rate of a column (Ve) must not exceed its flooding rate; cf. expressions (15). The flooding rate of a column is a function of its flooding velocity, ve, the crosssectional area of the column, Se (which is a function of the column diameter, De), and the liquid and vapor densities, FL and FG, respectively, as indicated in inequality (15). The flooding velocity of the column is approximated by a constant function of its tray spacing as indicated in eq 16. The matching of tasks with

Ve e FGveSe

(Se ) πDe2/4)

(15)

x

FL - FG FG (K ) 0.24 for tray spacing ∼ 18 in.) (16)

ve ) K

processing equipment must also satisfy task-specific feasibility constraints. Feasibility constraints disqualify the application of certain equipment types for a specific task [expression (12)]. For distillation columns, feasibility constraints ensure the minimum column height, He, for performing a given separation as indicated in inequality (17). In our program, infeasible matches are examined and enforced a priori with the help of integer variables nΤ,e set to zero.

Nopβ e Ηe

(β ) tray spacing ∼ 18 in.) (17)

If a given operation, T, requires more units than available at the site, investment decisions for purchasing new equipment can be taken; cf. expressions (6). Alternately, the program may abandon the treatment task, T, and swap to a new operating policy by assigning a zero to the binary decision variable associated with that task, xk,T. In that case, all of the associated equipment matches also become zero via the logic of inequalities (10) and (11), i.e., nT,e ) 0. 3.3. Accommodating Future Regulatory Changes: Regulatory Forecasts. Drug manufacturing sites have to ensure the desired production levels of the company’s product portfolio. Severe business setbacks could result from the manufacturing site’s inability to meet the desired demand. Production shortfalls could occur for two major reasons: (i) bottlenecks in the production inventory and (ii) problems in handling unavoidable wastes. This paper focuses on inadequate waste management, because this issue has not been addressed systematically before. In particular, an inability to comply with regulation or permits governing the manufacturing site emissions may impede necessary production expansions. The problem of meeting environmental standards is aggravated by the everlasting cycles of new environmental legislation as well as the long duration in obtaining permits for new abatement technology. As an example consider the acquisition of an installation permit for a new waste incinerator, which entails a legal procedure of several years. We propose to protect desired manufacturing plans from these uncertainties by considering current emission levels and anticipating future regulatory changes in the facilities planning. The regulatory forecast reflects expected or probable changes in the environmental regulations. It is worth mentioning that site-specific regulations for landfills, wastewater disposal, or air emissions vary from one geographic location to another and can change over time. Future regulations may limit total volatile organic emissions35 and CO2 emissions and ban entire treatment options such as prohibiting incineration of chlorinated compounds or landfill of any pharmaceutical residue. In our approach, peak limits and total emission caps affecting plant-wide discharges are enforced via emission inequality constraints; see inequality (13). They denote hard site-specific regulatory constraints on CO2, volatile organic carbons (VOCs), hazardous air pollutants (HAPs), landfill, etc., which are subject to change over time as anticipated by the regulatory forecast. The matrix, E(t), computed in the superstructure generation step, captures the amount of final discharges of all emission types caused by terminal waste residuals. Generation of Problem Scenarios. Regulatory forecasts can be generated using qualitative techniques described above. A pairing of different business and regulatory scenarios leads to a site management scenario matrix. Figure 2 proposes nine possible scenarios for the future development of the business and regulatory context. For the subsequent multiperiod optimization, we recommend to treat each projected scenario as an independent problem. It is also conceivable to consider different versions of the regulatory database R(t) in each planning period to generate transient superstructures of possible recycle and treatment options.

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for recovery and treatment operations expressed by the binary variables, x(t), in the superstructure generation phase. The logical path constraints of eq 14 enforce the correct connectivity of the flowsheets, as was found in the superstructure generation stage of the combinatorial synthesis methodology. A numerical example for independent validation of our proposed methodology is given in the appendix. 4. Industrial Problem (Case Study A)

Figure 2. Schematic of the forecast scenario matrix.

Figure 3. Waste load forecast for eight streams, W1-W8, obtained from business and market forecasts.

Path Connectivity Constraints. The design variables in the MILP of eqs 1-14 also include the choices

We would like to study the evolution of the particular multipurpose batch manufacturing plant for the eight waste categories presented in part 1 of this series,1 W1W8. These effluents coming from a synthetic organic pharmaceutical plant include extraction and wash solvents. All but one blend are organic mixtures with high recovery potential (W1-W6 and W8); effluent W7 is aqueous contaminated with trace organics. The superstructure of treatment and recovery options for these eight waste blends implicitly embeds a total of 16 000 different waste treatment policies, as was shown in detail earlier.1,2 Waste Forecast. We infer the waste forecast of expected discharges and compositions from the production plan over the next 5 years. Figure 3 visualizes the waste forecast for these eight waste categories over 5 years. The effluents W1, W2, and W8 stem from manufacturing campaigns of products with high market demand soon. Therefore, strong growth in their production levels together with increased waste discharges are expected. The trends for the remaining products and their associated waste categories exhibit slow increments. Plant Model. The plant model of Figure 4 schematically depicts the initial equipment inventory at the fictitious site A. The facility of our example is equipped with process units similar to those available at real

Figure 4. Initial plant inventory model for a hypothetical industrial site at time t ) 0.

4778 Ind. Eng. Chem. Res., Vol. 42, No. 20, 2003 Table 5. Pretreatment of a Wastewater Plant: Standard Ion Exchanger Size and Their Purchase Price type

capacity (m3/h)

installed cost ($)

ion-1 ion-2 ion-3 ion-4

50 90 200 540

400 000 790 000 1 000 000 1 250 000

a Installed cost estimated using MATCHE Software38 (cost based on year 2000).

industrial sites. The site’s inventory consists of a medium-sized incineration unit, a solvent recovery plant, a wastewater treatment facility, and a landfill. Special effluents or excess loads not treatable on site may also be sent to specialized off-site facilities. Large manufacturing sites may also have their own hazardous landfill. A brief description of recovery and treatment facilities used in the industrial case study follows. More details on the selection of recovery and treatment processes and capital cost estimation are discussed elsewhere.27 Incineration Plant. The incineration plant may consist of different types of incinerators, e.g., hazardous waste incinerator, liquid destructors, rotary kilns, etc.36 Incinerators are often equipped with ancillary environmental control technology such as electrostatic precipitators, dust collectors, wet scrubbers, venturi scrubbers, etc. For our plant inventory model, we consider one hazardous waste incinerator of the liquid destructor type with a thermal rating of 50 MBtu/h equipped with a wet scrubber (capacity ∼ 10 000 cfm) for emission control. Solvent Recovery Facility. Pharmaceutical manufacturing leads to large amounts of liquid organic byproduct streams. Trapped solvents can be recovered in dedicated solvent recovery plants consisting of batch or continuous columns of various heights and diameters. Our facility houses 12 continuous distillation columns with heights ranging from 22 to 200 ft (14-100 trays) and diameters between 12 and 36 in. Tables 3 and 4 compile relevant properties of the distillation towers and thermal incinerators. The purchase costs for the different equipment types are approximated by empirical cost correlations37,38 or commercial cost estimation software, e.g., MATCHES39 and DORT.40,41 For industrial applications, these data can easily be replaced with corporate-specific models or vendor quotes, e.g., SEC Heat Exchanger.42 Wastewater Treatment Facility. Wastewater treatment facilities (WWT) typically provide equalization/ diversion basins, neutralization (pH adjustment), anaerobic and/or aerobic digestion, clarifiers/filters, etc. Wastewaters often contain salts alongside organics. Therefore, ion exchange/reverse osmosis units are common as pretreatment units.36 Generally, the capital and operating costs of reverse osmosis are higher than those of ion exchange. Unless regional regulations for chemical and wastewater disposal require otherwise, ion exchange, being cheaper, is usually opted for by industry. A sample of commercially available ion-exchange modules and their installed costs are listed in Table 5. Commercial wastewater plants can handle between 5 and 40 million gal/day.43 Most wastewater treatment plants are custom-built; therefore, total capacities are site-specific. Table 6 gives the incremental investment and operating cost for a hazardous WWT plant.44 The operating expenditure of WWT plants correlates with hydraulic and organic waste loads send to the facility.

Figure 5. Cumulative annualized cost of strategies S1-S4. S1 ) unconstrained strategy (scenario WM1); S2 ) strategy with budget limitations (scenario WM2); S3 ) strategy assuming a limit on CO2 (scenario WM3); S4 ) strategy with both budget limitations and new CO2 regulations (scenario WM4). Table 6. Incremental Investment and Operating Cost for Wastewater Treatment Plants44 wastewater load hydraulic (g/m) organic

incremental investment (1997 $)

incremental operating cost (1999 $)

3000 per g/m 6000 per lb of organic/h

$300/year per g/m $2000/year per lb of organic/h

In our hypothetical plant inventory model, we have considered a 5 million gal/day wastewater treatment facility with an ion-exchange unit (capacity ∼ 50 m3/h) for pretreatment. Off-Site Waste Treatment. Special effluents or overloads from a manufacturing campaign can be stored temporarily in tank farm before being directed to offsite treatment facilities. Off-site treatment costs greatly depend on the type of waste (e.g., solid or liquid, hazardous or nonhazardous), the nature of the hazardous constituents, and the geographical location (e.g., supply and demand for off-site waste treatment capacity). The specific cost for off-site treatment varies from site to site. Typical examples of specific off-site costs are listed in Table 2.44 The cost of transportation is a function of the nature of the waste (e.g., hazardous or nonhazardous) and the proximity to the off-site facility. However, off-site transporation costs generally add only 2-4¢/lb to the values of Table 2.44 Accordingly, we have neglected transportation cost. If transportation from one state to another is needed, new local regulation(s) may enforce additional taxes, which could surpass the actual cost of waste treatment. Capital Constraints. Investments for technological innovations or necessary plant expansions may be limited by the available budget. In our case study, we considered a total investment budget of $5M. Similarly, yearly expenditure limits can easily be incorporated into our methodology. Regulatory Forecast. In this case study, it is initially assumed that there are no limits on CO2 emissions. Furthermore, we want to study the possible impact of a future regulatory change that caps total CO2

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Figure 6. Case study A: operating policies π1 (partial view) and π2 of strategy S1. Original waste streams labeled W1-W8; treatment steps enumerated from T1 to T135; residual streams denoted as S3 to S106 (OnR ) on-site recycle; INC ) incineration; REU ) reuse; EVA ) evaporation; CON ) condensation; DRY ) drying; IonE ) ion exchange; SCR ) scrubber; WAO ) wet air oxidation; BIO ) biological treatment; LEA ) leaching; LF ) landfill; ATM ) atmosphere; SEW ) sewer). Table 7. Waste Management Scenarios Obtained from Business and Regulatory Forecasts (Case Study A) waste management scenarios

regulatory forecast

business forecast

WM1 WM2 WM3 WM4

no new regulations no new regulations CO2 emissions e 65 kton/year (from year 2) CO2 emissions e 65 kton/year (from year 2)

no capital constraint total capital investment e $5M no capital constraint total capital investment e $5M

emission of the site at 65 ktons/year. The new regulation of this fictitious scenario is assumed to take effect in 2 years. Waste Management Scenarios. Under the assumptions made above, we arrive at four waste management scenarios, WM1-WM4 tabulated in Table 7. The optimistic scenario, WM1, assumes no change in regulations and operates without capital expenditure constraints. Scenario WM4, the most constrained scenario, prepares for tighter regulations and a limited investment budget. 4.1. Minimum Cost Waste Management Strategy (without Limitations). In this case study the MILP of eqs 1-14 was solved using the plant-model A, the waste forecast information of Figure 3, and the waste management scenarios WM1 of Table 7. Candidate strategies were allowed to propose optimal treatment policies and investments for the first 5 years. To evaluate the long-term impact of each strategy, an economic horizon of 20 years with constant inputs after the fifth year was considered. Plant Operations. The optimal strategy, S1, without capital limitations or expected regulatory changes, achieves the lowest possible annualized cost (see Figure 5). This waste management strategy swaps operating policies in years 1, 2, and 5, respectively. The tasks of policy π0 mainly using incineration lead to high operating cost. In year 1, swapping to π1 exploits higher levels of solvent recovery including some difficult distillations for the organic solvent mixtures (W1 and W4). This highrecovery approach of π1 slightly reduces the operating cost in comparison to the first year (π0). Policy π2, in year 2, avoids the most difficult separations (i.e., recovery of benzene and ethylene dichloride from a ternary benzene-ethylene dichloride-toluene mixture through tasks T4a and T4b of Figure 6). Instead, it favors self-sustained incineration (i.e., task T5), leading to the lowest annual operating cost policy (π2). Figure 6 depicts the detailed recycle and treatment tasks of policy π2,

with the best tradeoff between solvent recovery and destructive treatment. It also compares the destructive treatment for the quaternary spent solvent mix (W1) with the complex and expensive recycle operations proposed by policy π1. Finally in year 5, strategy S1 reverts back to higher levels of solvent recycle (policy π1) because of the limited incinerator capacities. Equipment Investments. To realize the self-sustaining incineration task of policy π2, installation of a new incinerator with a capacity of 50 MBtu/h is proposed in year 2. The incineration choices of policy π2 could not be implemented in the first year because of the time needed for commissioning the new incinerator. Time delays for obtaining operation permits for special equipment types are incorporated into the proposed methodology via permit constraints. Strategy S1 suggests the purchase of new distillation columns in years 1 and 5 for recycling benzene and ethylene dichloride from effluent W1. Column investments are also needed to handle the growth in waste loads, e.g., one column of type C4 to handle increased discharge of W3 in policy π2. The total annual investment expenditures of strategy S1 are tabulated in Figure 5. A more detailed analysis of the task-equipment matches for the 5 years of operation of the site follows. Optimal Column Allocation. The MILP of problem A, eqs 1-14, also was solved for the optimal column allocation for all separation tasks shown in Figure 7 for the first two planning periods. It lists a total of 10 separation tasks (T1, T4, T32, T37, T48, T65, T86, T88, T91, and T108) corresponding to separation tasks found in the superstructure generation phase. Figure 7 also tabulates the number of trays and operating heights required for each distillation tower. Clearly, a column of type C1 with 15 trays can only support the separation task Τ37 requiring at least 13 trays for the chosen reflux. All columns of type C8 and C9 equipped with 130 trays can handle any of the solvent recovery tasks. However,

4780 Ind. Eng. Chem. Res., Vol. 42, No. 20, 2003

Figure 7. Optimal facility allocation for strategy S1: matching of separation tasks with distillation columns (years 0 and 1).

deployment of large distillation towers incurs higher maintenance costs and equipment charges. It should be noted that some separation tasks recycle intermediate boiling solvents. Accordingly, recovery of intermediate cuts requires a multicolumn sequence with separation subtasks, e.g., two columns for the separation of the intermediate boiler, acetonitrile, from a ternary acetone-propanol-acetonitrile mixture in task Τ108 (subtasks Τ108a and Τ108b). Figures 7 and 8 display the task-equipment allocation for each planning period. They also show that the columns assigned to a separation task in one period might be reassigned to another task in the subsequent period (e.g., task Τ91, for recycling propanol from a binary propanol-acetone mixture, executed in a column of type C5 in year 0 and a column of type C2 in year 1). Moreover, several columns may be used together to perform a single recycle operation (e.g., combined use of columns of type C5 and C6 for the task Τ108b to recycle of acetonitrile from the ternary acetone-acetonitrile-propanol solvent mixture, in year 1). New column investments are visualized by black dots in the year of purchase, e.g., one type C1 column in year 1. Changes in operating policies may replace solvent recovery tasks in favor of a destructive treatment, e.g., incineration. Separation tasks not selected in the

current operating policy are marked in gray. Figure 8 illustrates the column allocation for the rest of the planning horizon from years 2 to 5. It is interesting to observe that the column allocation may change even with fixed plant operations. This practice can be explained by the need for larger towers in response to increasing waste discharges. The task-equipment matching in the proposed methodology should not be confused with a detailed scheduling problem. Our approach ensures availability of a sufficient number and type of processing equipment for a budget year. It does not attempt to address a detailed scheduling of the separation tasks or the units they occupy. Alternative Strategies. Strategy S1 deploys optimal operating policy adjustments in years 1, 2, and 5 to maximize benefits from solvent recovery while avoiding excessive expenses associated with difficult separations. An alternative strategy, SSupOpt, with suboptimal recovery and treatment policies is also investigated. The inferior strategy commences with maximum incineration (π0) followed by most aggressive recycling from year 1 onward (π1). Figure 9 compares the incinerator usage in strategy S1 and an inferior strategy, SSupOpt. Although the suboptimal strategy deploys more recycle steps, it still requires the new incinerator investment proposed

Ind. Eng. Chem. Res., Vol. 42, No. 20, 2003 4781

Figure 8. Optimal facility allocation for strategy S1: matching of separation tasks with distillation columns (years 2 and 5).

in the cost-optimal strategy S1. Hence, it cannot avoid the associated capital cost; at the same it runs this new incinerator only at partial capacity. In effect, it operates approximately 2.6% more expensively than the minimum cost strategy S1. 4.2. Optimal Waste Management Strategies with Capital Limitations and Changes in Regulations. According to the scenario matrix of Table 7, four situations responding to projected expectations of future plant requirements are of interest. Consequently, we computed four optimal strategies, S1-S4, each one suggesting appropriate operational and investment decisions for the particular future scenario. Figure 5 compares the cumulative annualized cost of four optimal strategies, S1-S4, resulting from applying the MILP of eqs 1-14 to the scenarios WM1-WM4 respectively. The investment decisions for new columns and incinerators for each of these strategies are enumerated in Table 8. Strategies under Budget Restrictions. Strategy S2, constrained within a $5M investment budget, affords fewer investment decisions than S1. It spreads out the investment decisions over the entire planning horizon such that the net present value of capital investments stays within the limited budget (cf. Figure 5). Strategy

S2 swaps operating policies after the first year (policy π0 f π2) and then again in year 3 (policy π2 f π1), where it remains until the end of the planning period. Although S2 deploys similar operating policies, its total cost is twice as high as that of strategy S1. The cost increase is attributable to the capital limitations of S2, leading to a suboptimal investment schedule with a delay in the necessary plant expansion. Costly off-site treatment and their associated cost penalties are the result in this scenario S2. Strategies for Tighter Regulations. Strategy S3 answers to new environmental laws limiting the site’s total CO2 emission at 65 ktons/year taking effect in year 2. Because of the CO2 emission limit, strategy S3 cannot maintain the cheapest operations using self-sustaining incineration (π2). It maintains the high recycle policy π1 until year 3 and then adopts new policy π3 in year 4. This new policy π3 deploys more separation steps than any of the earlier policies (π0-π2), leading to the highest operating cost. Figure 10 depicts the recovery and treatment task network of operating policies π3. A comparison of Figures 6 and 10 reveals why policy π3 produces less CO2 emissions. It deploys difficult separations such as

4782 Ind. Eng. Chem. Res., Vol. 42, No. 20, 2003 Table 8. Number of New Equipments Purchases for Strategies S1-S3 of Case Study A and Strategies S5-S7 of Case Study B (C1-C9, Distillation Columns; I1-I4; Incinerators) strategy S1 period (year)

strategy S2 period (year)

strategy S3 period (year)

equipment

1

2

3

4

5

1

2

3

4

5

1

2

3

4

5

C1 (22 ft × 12 in.) C2 (35 ft × 12 in.) C3 (82 ft × 12 in.) C4 (25 ft × 12 in.) C5 (35 ft × 18 in.) C6 (82 ft × 18 in.) C7 (82 ft × 24 in.) C8 (200 ft × 24 in.) C9 (200 ft × 36 in.) I1 (120 MBtu/h) I2 (80 MBtu/h) I3 (50 MBtu/h) I4 (30 MBtu/h)

0 1 0 0 0 2 1 0 1 0 0 0 0

0 0 0 0 0 0 0 0 0 0 0 1 0

0 0 0 0 0 0 0 0 0 0 0 0 0

0 0 0 0 0 0 0 0 0 0 0 0 0

0 0 0 0 0 1 2 0 0 0 0 0 0

0 0 0 0 0 1 0 0 0 0 0 0 0

0 0 0 0 0 1 0 0 0 0 0 0 1

0 0 0 0 0 0 1 0 0 0 0 0 0

0 0 0 0 0 1 2 0 1 0 0 0 0

0 1 0 0 0 0 0 0 0 0 0 0 0

0 0 0 0 0 3 1 0 1 0 0 0 0

0 0 0 0 0 0 1 0 0 0 0 1 0

0 1 0 0 0 0 0 0 0 0 0 0 0

0 0 0 0 0 0 1 0 0 0 0 1 0

0 0 0 0 0 0 0 0 0 0 0 0 0

strategy S5 period (year)

strategy S6 period (year)

strategy S7 period (year)

equipment

1

2

3

4

5

1

2

3

4

5

1

2

3

4

5

C1 (22 ft × 12 in.) C2 (35 ft × 12 in.) C3 (82 ft × 12 in.) C4 (25 ft × 12 in.) C5 (35 ft × 18 in.) C6 (82 ft × 18 in.) C7 (82 ft × 24 in.) C8 (200 ft × 24 in.) C9 (200 ft × 36 in.) I1 (120 MBtu/h) I2 (80 MBtu/h) I3 (50 MBtu/h) I4 (30 MBtu/h)

0 0 0 0 0 2 2 0 1 0 0 0 0

0 0 0 0 0 0 0 0 0 0 0 1 0

0 0 0 0 0 0 0 0 0 0 0 0 0

0 0 0 1 0 1 0 0 1 0 0 0 0

0 0 0 0 0 0 0 1 0 0 0 0 0

0 0 0 0 0 3 2 0 1 0 0 0 0

0 0 0 0 0 0 0 0 0 0 1 0 0

0 0 0 0 0 0 0 0 0 0 0 0 0

0 0 0 0 0 0 0 0 0 0 0 0 0

0 0 0 0 0 0 0 0 0 0 0 0 0

0 0 0 0 0 3 2 0 1 0 0 0 0

0 0 0 0 0 0 0 0 0 0 0 1 0

0 0 0 0 0 0 0 0 0 0 0 0 0

0 1 0 0 0 0 0 0 0 0 0 1 0

0 0 0 0 0 1 3 0 0 0 0 0 0

posed planning methodology rendered site waste management strategies capable of addressing regulatory changes, but at almost twice the cost.

Figure 9. Total resource usage for two strategies S1 and SSubOpt. Strategy S1 allows policy adjustments to avoid difficult separation in years 1, 2, and 5 while strategy SSubOpt does not allow any policy adjustments after year 1.

recycling acetone (S24) from a methanol-acetone-water mixture (W2) by leaching (T31). The subsequent incineration step, T35, emits less CO2 (S29) than the other policies that burned W2. However, another incinerator is required to handle the excess water introduced by the leaching operation (T35 in Figure 10). This strategy also implements more recycle steps than strategy S1. The new solvent recycle tasks are made possible by capital investments of $2.27M for a new solvent recovery plant with five additional columns (see Table 8). The total investment for new equipment including an incinerator and distillation columns amounts to $13.12M. The total cost of strategy S3 amounts to $149.17M, which is twice as expensive as that of strategy S1 (see Figure 5). We conclude that, in this case study, the anticipation of future regulatory changes combined with the pro-

Strategies for Tighter Regulations and Capital Limitations. Waste management strategy S4 optimal under the scenario with capital limitations and a CO2 emission limit (WM4) amounts to $454.48M. Its total annualized cost is approximately 3 times higher than the most expensive of the other three strategies. We conclude that a new regulation limiting the total CO2 emissions at the site would lead to extremely high manufacturing cost unless considerable investments are made. Therefore, it would be very expensive for the manufacturer to attain the desired degree of pollution prevention required by the new regulation without plant expansion. Figure 11 compares the total yearly CO2 emissions for the four strategies, S1-S4, for the entire planning horizon. Both strategies S1 and S2 would violate the new regulation threshold after year 2. Strategies S3 and S4 meet the new regulation constraint but incur higher operating costs and investment expenditures. Task Distribution of Strategies. Figure 12 breaks up the distribution of treatment and recovery tasks for the three strategies S1-S3 within the 5 years of the planning horizon. The strategy with budget constraints S2 relies heavily on off-site treatment for effluents that cannot be handled by the site’s incinerators. Instead of expensive investments for a high-capacity incinerator of 50 MBtu/h, it opts for purchasing less costly separation columns and engages in more solvent recovery. We also recognize that strategy S3 by addressing the CO2 regulation employs the largest degree of solvent recovery to bring down total CO2 emissions. All three strate-

Ind. Eng. Chem. Res., Vol. 42, No. 20, 2003 4783

Figure 10. Case study A: operating policy π3. Original waste streams labeled W1-W8; treatment steps enumerated from T1 to T135; residual streams denoted as S3-S106 (OnR ) on-site recycle; INC ) incineration; REU ) reuse; EVA ) evaporation; CON ) condensation; DRY ) drying; IonE ) ion exchange; SCR ) scrubber; WAO ) wet air oxidation; BIO ) biological treatment; LEA ) leaching; LF ) landfill; ATM ) atmosphere; SEW ) sewer).

gies S1-S3 use up identical levels of landfill space because landfill is chosen as the treatment option for residuals of stream W7 (T130) and W8 (T134). 5. Uncertainty in the Forecasts: Stochastic Multiperiod Planning The model presented above assumes perfect information of future waste management scenarios. Because of the uncertainty inherent to future events, it is desirable to quantify the variability associated with the predictions. For accommodating uncertainty in the waste forecasts, we propose the following probabilistic model: (i) We allow for variations around a nominal value according to a normal distribution given its mean, µ, and standard deviation, σ. Forecasting techniques are used to predict the mean of the future waste loads. The uncertainty in the forecasts is related to the standard deviation. The higher degree of uncertainty associated with the forecasts of more distant events is modeled via an increasing relative uncertainty factor. It is

Figure 11. Yearly CO2 emissions for strategies S1-S4. S1 ) unconstrained strategy; S2 ) strategy with budget limitations; S3 ) strategy assuming a limit on CO2 emissions; S4 ) strategy with both budget limitations and new CO2 regulations.

defined as the ratio of the variability of the waste load, σ, over its mean value, µ. Accordingly, the uncertainty attributed to the waste scenario forecasts increases monotonically in time.

4784 Ind. Eng. Chem. Res., Vol. 42, No. 20, 2003

Figure 12. Distribution of tasks for strategies S1-S3 for a planning horizon of 5 years. S1 ) unconstrained strategy; S2 ) strategy with budget limitations; S3 ) strategy assuming a limit on CO2 emissions.

(ii) To maintain mathematical tractability, we deploy sampling techniques discussed in part 2 of this series.2 MCS and Latin Hypercube sampling are used to model the uncertain space for each of the time periods. Joint probability distributions of the discrete waste forecast scenarios, Ws(t), are obtained by sampling the uncertain space of the individual waste load distribution in each of the planning periods. Necessary sample sizes were determined by the technique of bootstrapping.2,45 The multiperiod stochastic program of problem B characterizes the long-term planning problem considering uncertainty in the waste forecast. The problem emerges from the deterministic version presented in section 3, when replacing all deterministic cost and waste load data with their respective probabilistic values. The approach to attain discretized waste scenarios by the dynamic linearization technique was presented in detail in part 2 of this series.2 In stochastic multiperiod formulation of problem B, the objective (1b) includes the expected net present operating cost that can be obtained as the probabilistic average of the operating cost in the individual waste forecast scenarios, Ws(t) (eq 2b). The objective also includes deterministic models for net present capital cost (eq 5) and maintenance cost (eq 9). The capital expenditures are given deterministically, because investment decisions made in any time period are discrete, and so is the maintenance cost. We also enforce emission limitations for all uncertain scenarios, emax(t). The total emission caused by any scenario E[Ws(t)] must satisfy all thresholds emax(t) active in a given time period [inequality (13b)]. Unlike the environmental regulations, capacity constraints [equality (3b) and inequality (4b)] are not considered hard. Capacity constraints may be violated at the expense of a specific excess load penalty, µT. The expected value of these penalty costs can be obtained by computing the scenario-dependent capacity violations, pT; cf. expressions (3b) and (4b).

Expressions (6)-(8) of problem A model discrete investment decisions for this stochastic program as well. The same holds true for the equipment matching constraints of (10)-(12).

Problem B: Min

x(t),ne(t),nT,e(t), pT[Ws(t),t],qT[Ws(t),t]

NPC ) Exp{NPCR} + NPCβ + NPCγ (1b) n

Exp{NPCR} )

∑(1 + r) ∑ P(W (t))c(W (t),t)‚x(t) + -t

t)0

s

s

s∈M(t)

n

∑(µ p (W (t),t)) s

(2b)

T T

t)1

s dT(Ws(t),t)‚x ) dmax T (t) + qT(W (t),t);

∀ T ∈ IT (3b)

pT(Ws(t),t) g qT(Ws(t),t); pT(Ws(t),t) g 0;∀ T ∈ IT E(Ws(t),t)‚x(t) e emax(t)

(4b)

(13b)

and eqs 5, 6-12, and 14 of problem A. 5.1. Case Study B. This case study examines the impact of different degrees of uncertainty in these projections on the planning strategies. Accordingly, we composed a waste scenario matrix with three waste management scenarios with increasing relative uncertainty, WM5-WM7. Hence, the forecasts of WM5 are considered to be relatively accurate, while the quality of the predictions of WM7 have a relative uncertainty of 30%, 6 times the degree in WM5. The mean values of expected waste discharges in all three scenarios are identical and equal to those given by the forecasts of

Ind. Eng. Chem. Res., Vol. 42, No. 20, 2003 4785

Figure 13. Cost comparisons of uncertain waste management strategies (S4-S6) with S1. S1 ) strategy with no uncertainty (scenario WM1); S5 ) strategy with low relative uncertainty (scenario WM5); S6 ) strategy with medium relative uncertainty (scenario WM6); S7 ) strategy with high relative uncertainty (scenario WM7). Table 9. Waste Management Scenarios for Uncertainty in Forecasts (Case Study B) waste management scenarios

year 1

WM5 WM6 WM7

1 3 5

% relative uncertainty (σ/µ) year 2 year 3 year 4 year 5 2 5 10

3 9 15

4 12 20

5 15 30

Figure 3. Table 9 lists the relative uncertainty of the three waste management scenarios WM5-WM7. For scenario WM5, the relative uncertainty increases from 1% in year 1 to 5% in year 5, while for scenario WM7, it increases from 5 to 30% over the planning horizon. A total of 500 Monte Carlo samples were used to represent the uncertain space. The necessary sample size was determined by bootstrapping.2,45 Results. The stochastic MILP formulation of problem B was used to solve for optimal strategies under uncertainty for the three uncertain scenarios, WM5WM7. The cost values of the resulting waste management strategies S5-S7 are shown in Figure 13. The analysis reveals high capital investments required by strategy S7, which is associated with the most uncertain scenario WM7 (cf. strategies S1 and S7 in Figure 13). Clearly, high uncertainty in waste loads forced the purchase of larger units to handle large variations from the mean (e.g., very expensive incinerators). Despite higher investment costs to handle peak loads, the “expected” operating costs were hardly affected. In this example, the total annualized cost for a 20-year economic horizon was not sensitive to different degrees of uncertainty, as indicated in Figure 13. The expected operating costs for all scenarios are roughly the same because the mean discharges are identical. Hence, the total expected cost does not differ more than 10% because of differences in annualized capital investments. 6. Conclusions/Significance In this paper, we have proposed a novel multiperiod decision-making framework to obtain optimal long-term waste management strategies. Industrial case studies highlighted the tradeoffs in the proposed planning problem. Mixed-integer optimization programs are

deployed for finding long-term plant-wide operating policies together with optimal capital investment decisions for new processing equipment. The effect of future regulatory changes on plant operation has also been discussed. Using our computer-aided methodology, decision makers can examine a variety of different business and regulatory scenarios and arrive at plant-wide optimal strategies with little manual effort. Moreover, the proposed methodology with yearly production and regulatory updates can be used as a closed-loop operation and facility planning tool before actual investments are made. The plant management problem was cast into a simple MILP formulation and solved within execution times of less than 10 CPU min using the reduced branch and bound algorithm of the GAMS-CPLEX solver46 on a 1.6 GHz Pentium IV PC. This paper also concludes the third part of this series devoted to a comprehensive methodology for automatic synthesis of recovery and treatment options. Part 1 demonstrated finding the best tradeoff among conflicting objectives. Part 2 advanced the combinatorial process synthesis methodology to account for uncertainty in the waste loads. In this third paper, we proposed a multiperiod operations model with capacity expansion. The fundamental innovation in this series lies in the development and proof of the concept of a novel combinatorial process synthesis methodology. It deployed a constructive planning algorithm for dynamically and fully automatically building a superstructure of process tasks for recovering trapped raw materials or destroying offensive compounds through physical or chemical transformations. This superstructure, an acyclic directed graph, incorporated all alternative flowsheets composed of an ordered sequence of reaction and separation tasks. Its node information was used to automatically generate optimization problems for deterministic and stochastic versions of a plant-wide design problem. The successful application of the combinatorial process synthesis methodology holds high promise for more challenging computer-aided synthesis tasks such as the design of pharmaceutical batch recipes, synthesis of total heat-integrated refinery

4786 Ind. Eng. Chem. Res., Vol. 42, No. 20, 2003

operations, and fully automated synthesis of continuous chemical flowsheets.

Acknowledgment is made to the donors of the Petroleum Research Fund, administered by the ACS, for partial support of this research (ACS-PRF 35702-G9). A.C. is grateful to the Institute of Environmental Science and Policy at University of Illinois, Chicago, for providing financial support through the Environmental Manufacturing (EvMM) Fellowship. Provision of the optimization routines by GAMS Development Corp. is gratefully acknowledged. Appendix. Long-Term Deterministic Planning: A Numerical Example This appendix demonstrates an example providing all numerical details to reproduce the results of this paper. The objective in the test example is to find an optimal treatment strategy for the two waste streams of Figure 14 for a planning horizon of 3 years and an economic horizon of 10 years. The superstructure found with the methods described in part 1 of this series1 includes 15 treatment options in 6 different waste treatment policies. The MILP of eqs A1-A14 represents the detailed mathematical problem formulation for this example. The initial plant model, available equipment inventory, and assumed values of the waste forecasts for each time period are listed after eq A14. Min

NPC ) NPCR + NPCβ + NPCγ

3

NPCR )

(

∑(1 + r) ∑ c -t

∀xi,j

t)0

+

i,j

∂ci,j ∂wi

(A1)

∑(1 + r)

-t

)

{wi(t) - wi(0)} xi,j(t)

µTpT(t)

t)0 10

+

(

∑(1 + r) ∑ c -t

∀xi,j

t)4

i,j

+

)

∂ci,j

(wi(3) - wi(0)) xi,j(3)

∂wi

10

+

∑(1 + r)

-t

µTpT(3)

(A2)

t)4

3

NPCβ )

∑(1 + r) (∑∆n (t) C ) -t

e

dT(t) )

∑

e

(A3)

∀e

t)0

Fi,jwi(t) xi,j(t);

∀ T ∈ INC

(A4)

∀xi,j⊂INC

pT(t) g (dT(t) -

∑n (t) S ); e

e

∀e

pINC(t) g 0 (∀ T ∈ INC; ∀ e ∈ Incinerators)

(A5)

(Ri,j + 1)γi,jwi(t) xi,j(t) e

∑

nT,e(t) Se (∀ T∈ OnR; ∀ e ∈ Columns)

(A6)

∀T∈OnR

∑

(A9)

(A10)

3

NPCγ )

∑

ωT(1+r)-tnT,e(t) Ce (∀ T ∈ OnR; ∀ e ∈ Columns)

(A11)

t)0

x1,1(t) + x1,2(t) + x1,3(t) ) 1 x1,1(t) ) x1,4(t) ) x1,5(t) ) x1,9(t) ) x1,10(t); x1,3(t) ) x1,8(t) x2,1(t) + x2,2(t) ) 1;

(A12) x1,2(t) ) x1,6(t) ) x1,7(t); (A13)

x2,1(t) ) x2,3(t) ) x2,4(t);

x2,2(t) ) x2,5(t)

(A14)

with Waste forecast (t ) 1-3): w1(t) ) {11.00, 12.00, 13.00} × 106 kg;w1(t)0) ) 10.00 × 106 kg; w2(t) ) {7.50, 15.00, 12.50} × 106 kg; w2(t)0) ) 5.00 × 106 kg Feasibility constraints: Columns C2 and C3 are infeasible for separation task x2,1 (i.e., nT,e ) 0; T ) x2,1; e ) C2, C3) Constant parameters: r ) 0.10, µT ) 0.77, ωΤ ) 0.001 Plant model: e ) Incinerator {I1, I2, I3}; Columns {C1, C2, C3} Incinerator: types ) {I1, I2, I3}; Se ) {54.43, 34.02, 20.41} × 106 kg, Ce ) ${7.35, 4.93, 3.19} × 106; ne(0) ) {0, 0, 1} Columns: types ) {C1, C2, C3}; Se ) {18.90, 10.63, 4.72} × 106 kg; Ce ) ${0.716, 0.321, 0.184} × 106; ne(0) ) {0, 0, 1}

3

+

∆ne(0) ) 0 (∀ e ∈ Incinerators & Columns) ∆ne(1) ) 0 [∀ e ∈ Incinerators (permit constraint)]

Acknowledgment

xi,j(t),ne(t), nT,e(t),pT(t)

∆ne(t) ) ne(t) - ne(t - 1);

nT,e(t) e ne(t) (∀ T ∈ OnR; ∀ e ∈ Columns)

(A7)

nT,e(t) ) 0 (∀ T ∈ OnR; ∀ e ∈ Infeasible Columns)

(A8)

∀xi,j⊂OnR

This MILP formulation has a total of 36 integer decision variables in each time period of the planning horizon (i.e., years 0-3): (i) 15 binary decision variables, xi,j(t), denoting the selection of treatment options, (ii) 6 integer variables, ∆ne(t), for incremental investments in three column types C1-C3 and three incinerator types I1-I3, (iii) 6 integer variables, ne(t), for recording the plant infrastructure in each time period, and (iv) 9 integer variable, nT,e(t), matching the three separation tasks (x1,2, x1,4, and x2,1) with columns C1-C3. The feasibility constraint (A8) renders column types C2 and C3 infeasible for the difficult separation between acetone and methanol (task x2,1), thereby reducing the number of task-equipment matching variables, nT,e(t), to 7. For each year of the planning horizon (i.e., time periods 0-3), the program chooses an optimal operating policy from among the six different treatment policies in the superstructure. Therefore, the total number of structurally different waste management strategies embedded in this problem is 64 ) 1296. Clearly, the space of investment decisions is continuous. Table 10 lists all constant parameters of the example that were computed by dynamic linearization of mass and energy balances during the superstructure generation phase. Results. Table 11 summarizes the treatment policies and investment decisions of the optimal waste management strategy obtained for this small example. The

Ind. Eng. Chem. Res., Vol. 42, No. 20, 2003 4787 Table 10. Constant Parameters Obtained from Dynamic Linearization of Mass and Energy Balances xi,j

ci,j

∂ci,j/∂wi

Fi,j

γi,j

Ri,j

xi,j

ci,j

∂ci,j/∂wi

Fi,j

γi,j

Ri,j

x1,1 x1,2 x1,3 x1,4 x1,5 x1,6 x1,7

0 0.730 3.000 0.290 7.632 -5.759 3.833

1.000 0.007 0.030 0.003 0.076 -0.068 0.018

0 0 1.000 0 0.620 0 0

0 0.500 0 0.946 0 0 0

0 1.177 0 0.225 0 0 0

x1,8 x1,9 x1,10 x2,1 x2,2 x2,3 x2,4

0 -3.398 0 5.936 3 -2.679 -1.167

0 -0.144 0 0.129 0.030 -0.107 -0.052

0 0 0 0 1.000 0 0

0 0 0 0.500 0 0 0

0 0 0 7.835 0 0 0

Table 11. Optimal Waste Management Strategy for Two Waste Streams (Planning Horizon ) 3 years, Economic Horizon ) 10 years) time (yr)

operating policy

0 1

π0 π1

2 3

π2 π3

net present operating cost, NPCR net present capital investments, NPCβ net present maintenance cost, NPCγ NPC

investment schedule

treatment paths W1{x1,3 - x1,8}; W2{x2,2 - x2,5} W1{x1,1 - x1,4 - x1,5 - x1,9 - x1,10}; W2{x2,2 - x2,5} W1{x1,2- x1,6 - x1,7}; W2{x2,2 - x2,5} W1{x1,1 - x1,4 - x1,5 - x1,9 - x1,10}; W2{x2,1 - x2,3 - x2,4}

one C2 column three C3 columns

operating cost ($)

capital cost ($)

11.42 10 011.42

321 000

214 011.42 -419 988.58

552 000

)11.42 + 10011.42(1 + 0.1)-1 + 21401.42(1 + 0.1)-2 -419998.58(1 + 0.1)-3 + 10 (1 + 0.1)-t ) $-1665760 -419998.58∑t)4 )321000(1 + 0.1)-1 + 522000(1 + 0.1)-3 ) $1905642 $2869 (as in eq A11) NPCR + NPCβ + NPCγ ) -1665760 + 1905642 + 2869 ) $242751

The NPC of the optimal waste management strategy is $242 751. This value includes a huge recycle benefit of approximately $1.66M for recovered solvents. Alternatively, a strategy with a budget limitation of $1.5M for the 3-year planning horizon (i.e., NPCβ e 1 500 000) would have a NPC of $902 372. Nomenclature Indices s ) waste scenario T ) task belonging to treatment type T e ) equipment type belonging to treatment technology T Sets Figure 14. Superstructure for two simple waste streams (DEC ) decantation, OnR ) on-site recycle, INC ) incineration, REU ) reuse, ATM ) atmosphere, and WWT ) wastewater treatment).

optimal waste management strategy adopts a different operating policy every year. In year 0, the operating policy, π0, incinerates both waste streams (i.e., x1,3 ) 1; x2,2 ) 1). In year 1, recovery of acetone in W1 is made possible by decantation (x1,1 ) 1), followed by distillation (x1,4 ) 1) and reuse of acetone (x1,9 ) 1). The distillation step requires the purchase of a column of type C2 [i.e., ∆nC2(1) ) 1]. The aqueous phase from the decanter is sent to an incinerator whose offgas goes to the atmosphere (i.e., x1,5 ) x1,9 ) 1). In year 2, the operating policy π2 recovers acetone directly by high-purity distillation (x1,2 ) 1). The aqueous bottoms is sent to a wastewater treatment facility. Both operating policies π1 and π2 incinerate waste stream W2 instead of performing the difficult separation between methanol and acetone. In year 3, the program chooses to invest in three additional distillation columns of type C1 [∆nC1(3) ) 3] and adopting a recycling policy, π3, for recovering acetone and methanol from waste mixture W2 instead of incineration (x2,1 ) 1). The last year’s treatment policy, π3, stays in vigor until the end of the 10-year planning horizon.

Ws ) waste load outcomes of a particular scenario, s IT ) index set of tasks belonging to treatment type T Le ) index set of equipment types Labels SS ) superstructure WΜ ) waste management scenario p ) single-period waste treatment policy S ) multiperiod waste management strategy Parameters M(t) ) sample size for a waste scenario in year t r ) rate of interest µT ) specific penalty for capacity constraint violation for treatment T Ce ) price of equipment e P(Ws) ) probability of waste scenario Ws Se ) size of equipment e ωT ) specific maintenance cost for treatment type T Variables xk,i ) binary decision variable corresponding to the ith treatment task of effluent k NPC ) total net present cost NPCR ) net present operating cost NPCβ ) net present capital investments NPCγ ) net present maintenance cost

4788 Ind. Eng. Chem. Res., Vol. 42, No. 20, 2003 qT ) slack for capacity constraints of treatment T pT ) capacity overload for treatment T ne(t) ) number of equipment of type e available in year t ∆ne(t) ) number of equipment of type e purchased in year t dmax ) maximum capacity of treatment T (tons/year) T ∆dT(t) ) capacity increment of treatment type T in year t nΤ,e(t) ) number of equipment, e, associated with a treatment task, T Vectors c ) operating cost x ) binary decision variables for treatment selection dT ) load of treatment task T (tons/year) emax ) emission thresholds (tons/year) Matrixes E ) matrix of terminal emissions

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Received for review December 30, 2002 Revised manuscript received June 10, 2003 Accepted June 12, 2003 IE0210614