Performing capital investment planning (CIP) is traditionally done using linear (LP) or nonlinear (NLP) models whereby a gamut of scenarios are generated and manually searched to make expand and/or install decisions. Though mixed-integer nonlinear (MINLP) solvers have made significant advancements, they are often slow for industrial expenditure optimizations. We propose a more tractable approach using mixed-integer linear (MILP) model and input-output (Leontief) models whereby the nonlinearities are approximated to linearized operations, activities, or modes in large-scaled flowsheet problems. To model the different types of CIP's known as revamping, retrofitting, and repairing, we unify the modeling by combining planning balances with the scheduling concepts of sequence-dependent changeovers to represent the construction, commission, and correction stages explicitly. Similar applications can be applied to process design synthesis, asset allocation and utilization, and turnaround and inspection scheduling. Two motivating examples illustrate the modeling, and a retrofit example and an oil-refinery investment planning are highlighted.

1. 1
Generalized Capital Investment Planning of Oil-
Refineries using MILP and Sequence-Dependent
Setups
Brenno C. Menezes,a,b Jeffrey D. Kelly,c,* Ignacio E. Grossmann,d Alkis Vazacopoulose
aRefining Optimization, PETROBRAS Headquarters, Rio de Janeiro, Brazil.
bCenter for Information, Automation and Mobility, Technological Research Institute, São Paulo,
Brazil.
cIndustrial Algorithms LLC., 15 St. Andrews Road, Toronto, Canada.
dChemical Engineering Department, Carnegie Mellon University, Pittsburgh, United States.
eIndustrial Algorithms LLC., 202 Parkway, Harrington Park, United States.
Oil-refinery production, Investment planning, Process design synthesis, Sequence-dependent
changeovers
Abstract
Due to quantity times quality nonlinear terms inherent in the oil-refining industry, performing
industrial-sized capital investment planning (CIP) in this field is traditionally done using linear

2. 2
(LP) or nonlinear (NLP) models whereby a gamut of scenarios are generated and manually
searched to make expand and/or install decisions. Though mixed-integer nonlinear (MINLP)
solvers have made significant advancements, they are often slow for large industrial applications
in optimization; hence, we propose a more tractable approach to solve the CIP problem using a
mixed-integer linear programming (MILP) model and input-output (Leontief) models, where the
nonlinearities are approximated to linearized operations, activities, or modes in large-scaled
flowsheet problems. To model the different types of CIP's known as revamping, retrofitting, and
repairing, we unify the modeling by combining planning balances with scheduling concepts of
sequence-dependent changeovers to represent the construction, commission, and correction
stages explicitly in similar applications such as process design synthesis, asset allocation and
utilization, and turnaround and inspection scheduling. Two motivating examples illustrate the
modeling, and a retrofit example and an oil-refinery investment planning problem are also
highlighted.
1. Introduction
In the oil-refining industry, physical and chemical processes separate and convert a mix of
hydrocarbon molecules known as crude-oil resulting in derivatives or intermediate fuels to be
treated and blended into final products such as gasoline, kerosene and diesel. As a result of this
complexity that varies in a three-dimensional quantity-logic-quality relationship, industrial-sized
models integrating phenomenological (blending, processing, separating) and procedural
(sequences, setups, startups) optimization in planning and scheduling problems can be very
difficult to solve in a full space MINLP. Therefore, solution strategies such as MILP + NLP
phenomenological decompositions (Mouret, Grossmann, and Pestiaux, 2009; Menezes, Kelly, and

3. 3
Grossmann, 2015) and MILP approximations, as stated in this paper, can be proposed to handle
such complicated models with reasonable accuracy. However, in the strategic investment
planning optimization to construct oil-refineries or oil and gas facilities, most methodologies are
still based on simulation of numerous scenarios, reducing the models to LP and NLP problems
where the set of material flows and operating conditions are optimized considering the selected
process design and logistics frameworks.
These trial-and-error or try-and-test methodologies are also true in industrial scheduling
problems commonly performed among the oil-refineries worldwide. Either commercial or home-
grown scheduling solutions rely on simulation of events or situations to test feasibility, where the
user is responsible for trying different decisions manually as well as keeping track of convoluted
decision-trees that did not work or prove feasible. A normal outcome is that the schedulers
abandon these solutions, and return to their simpler spreadsheets because of the exhausting
efforts to model, memorize and manage the numerous scenarios and update the modeling
premises and situations that are constantly changing. Unfortunately, this translates into inferior
scheduling decisions that are made, which results in reduced operating capacity and capability as
well as increasing the variability of the whole system. This will ultimately feedback into the
planning bounds or limits, which will further result in inferior planning decision-making (Kelly
and Mann, 2003; and Kelly and Zyngier, 2008a).
To optimize process design in oil-refineries, we propose an input-output or Leontief (Leontief,
1986) modeling, also found in generalized network-flow and convergent and divergent problems,
to allow all of the units, facilities, and equipment to be modeled both with multiple operations or
activities and with multiple inputs and outputs interconnected both upstream and downstream
forming a complex network, chain, or more appropriately an arbitrary superstructure. These can

4. 4
be easily represented in large-scale and sophisticated optimization problems using a new
modeling and solving platform called IMPL (Industrial Modeling and Programming Language)
from Industrial Algorithms, LLC. that is a flowsheet, fundamentals, and formula-based
environment. This platform has built-in facilities for network and equation constructions using
the unit-operation-port-state superstructure (UOPSS) formulation that is the cross-product of the
physical (units) and the procedural (operations) models or substructures (Kelly, 2004b; Kelly,
2005; Zyngier and Kelly, 2012). Additional details on UOPSS and IMPL’s configuration can be
found in section 4 with the aid of the supplementary material.
In contrast to the use of MINLP models to optimize nonlinear continuous and discrete variables
in a full range space, MILP input-output models vary in a set of modes defined by parameters
such as yields, rates, and sizes and circumvent some of the drawbacks of the MINLP models at
the expense of the linearization. These include providing good initial-values of the continuous
variables to avoid infeasibilities in the nonlinear programming sub-problems, and the difficulty
of even solving the root relaxation NLP node when the binary or discrete variables are treated as
continuous variables. Furthermore given the inherent non-convexities, solving an MINLP to
global optimality may become intractable for medium and large scale problems especially when
minimizing or maximizing a weak and degenerate objective function. MILP input-output
approaches do not suffer from these problems, although they can vary only for the points or
regions set in the modes, i.e., within a reduced space. Therefore, in general they provide only an
approximation to the original MINLP, albeit with much greater robustness and reliability, of at
least attaining good quantity and logic feasible solutions in a reasonable amount of time.
The proposed MILP formulation represents the selection of projects and their intrinsic stages to
improve or extend/enhance assets by capacity, capability, or overall facilities expansion in which

5. 5
admissible project schedules must obey certain constraints such as stage dependencies, product
demands, and other resource restrictions. We address the modeling of stages, activities, or tasks
explicitly in order to better predict the different types of capital investment planning activities
known as revamping, retrofitting (Madron, 1992), and repairing especially found in the
petroleum, petrochemicals, and oil and gas industries. We generalize or unify the modeling by
combining supply-chain production and inventory planning balances with the scheduling
concepts of sequence-dependent setups, switchovers or changeovers to represent the
construction, commission, and correction stages, in which required capital resources can be
defined by product demands to be matched or limit the number of projects to be approved. The
importance of the stages is that during their executions, e.g., the existing assets are totally or
partially shutdown, so that the plant production can be modified within its project execution time
windows. This to our knowledge, has not been addressed in the conventional capital investment
planning found in the process industry literature (Sahinidis, Grossmann, Fornari, and Chathrathi,
1989; Liu and Sahinidis, 1996; Iyer and Grosmann, 1998; Van den Heever and Grossmann,
1999; Jackson and Grossmann, 2002; Al-Qahtani and Elkamel, 2008, 2009; Menezes et al.,
2014a; Menezes, Kelly and Grossmann, 2015).
The proposed approach can be employed in similar applications to what is known as production
and process design synthesis, asset allocation and utilization, and turnaround and inspection
scheduling, where binary unit-operation-time decisions to invest (revamp), operate (retrofit),
maintain (repair), etc. over these different problems are similarly modeled considering stages,
sequencing, and selection of states or modes by configuring them in IMPL’s embedded
semantics that uses the discrete-time sequence-dependent setup modeling of Kelly and Zyngier
(2007). Two motivating examples describe the modeling including the IMPL configuration and

6. 6
model formulation of the motivating example 1 (see the supplementary material). A retrofit case
reproduced from Jackson and Grossmann (2002) and an investment planning of an oil-refinery
plant are given as examples.
2. Conventional Capital Investment Planning (CCIP) Model
Capital investment planning (CIP) involves anticipating the investment of future assets for long-
term or strategic financial studies. The result is a list of new and/or enhanced equipment to be
invested in and a plan for financing the projects into a timetable or timeline for their completion.
Financial accounting objectives for rate of return, payback period, net present value, profitability
index, breakeven analysis, among others, and along with economic details on project plans and
their in- and out-capital flows, the CIP’s managerial or technical outcomes are its design capacity
and capability.
The aforementioned process industry literature solving the CIP problem disregard the operational
or production yields alongside the stages of completion of the projects. We name these type of
problems as conventional capital investment planning. They in general consider overall mass or
volume balances of chemicals (raw material and final products) in a single production site with a
network of processes interconnected by material streams to determine capacity expansion of
existing units over time respecting expected market demand (Shah, Li and Ierapetritou, 2011).
The conventional capital investment (CCIP) model is given in Eqs. (1)-(7) (Sahinidis,
Grossmann, Fornari, and Chathrathi, 1989), where its variables are not included in the
nomenclature section, instead we detailed the variables of the generalized capital investment
planning (GCIP) approach given in further sections. The CCIP model in Eqs. (1)-(7) and its

7. 7
related variables are presented here only for the sake of comparison with the proposed GCIP
modeling.
𝑦 𝑢,𝑡 𝑄𝐸 𝑢
𝐿
≤ 𝑄𝐸 𝑢,𝑡 ≤ 𝑦 𝑢,𝑡 𝑄𝐸 𝑢
𝑈
∀ 𝑢, 𝑡 (1)
𝑄𝐶 𝑢,𝑡+1 = 𝑄𝐶 𝑢,𝑡 + 𝑄𝐸 𝑢,𝑡 ∀ 𝑢, 𝑡 (2)
𝑄𝐹𝑢,𝑡 ≤ 𝑄𝐶 𝑢,𝑡 ∀ 𝑢, 𝑡 (3)
𝑃𝑠,𝑡 + ∑ 𝑊𝑢 ,𝑠,𝑡
𝑢∈𝑈 𝑆𝑂
= 𝑆𝑠,𝑡 + ∑ 𝑊𝑢,𝑠,𝑡
𝑢∈𝑈 𝑆𝐼
∀ 𝑠, 𝑡 (4)
𝑊𝑢 ,𝑠,𝑡 = 𝑓𝑢,𝑠,𝑡 𝑄𝐹𝑢,𝑡 ∀ 𝑢 ∈ 𝑈𝑆𝑂, 𝑠, 𝑡 (5)
∑(𝛼 𝑢,𝑡 𝑄𝐸 𝑢,𝑡 + 𝛽 𝑢,𝑡 𝑦 𝑢,𝑡) ≤ 𝐶𝐼𝑡
𝑢
∀ 𝑡 (6)
max 𝑁𝑃𝑉 = ∑ ∑(𝑝𝑟𝑠,𝑡 𝑆𝑠,𝑡 − 𝑝𝑟𝑠 ,𝑡 𝑃𝑠,𝑡)
𝑡𝑠
− ∑ ∑ 𝑐 𝑢,𝑡 𝑄𝐹𝑢,𝑡
𝑡𝑢
− ∑∑(𝛼 𝑢,𝑡 𝑄𝐸 𝑢,𝑡 + 𝛽 𝑢,𝑡 𝑦 𝑢,𝑡)
𝑡𝑢
(7)
In the CCIP modeling, the capacity expansion QEu,t of a given unit u in a certain time t is limited,
or lower (𝑄𝐸 𝑢
𝐿
) and upper (𝑄𝐸 𝑢
𝑈
) bounded, by the semi-continuous constraint in Eq (1). Over the
time, the next capacity size QCu,t+1 of u is the sum of its current capacity QCu,t plus the capacity
expansion QEu,t permitted at t as shown in Eq. (2). The connection between the design and the
operational layers are given by Eq. (3), in which operational unit throughputs QFu,t are upper
bounded by their current capacity QCu,t. Eq. (4) represents the volume/mass balance of all type of
stream s considering production of some in units or sources of s (u USO) giving by product

8. 8
yields fu,s,t and throughputs QFu,t as in Eq.(5). The overall quantity balance of stream s in Eq. (4)
is completed by its purchases Ps,t and sales Ss,t, where Wu,s,t represents on the left hand, sources or
producers of s (u USO) and, on the right hand, its sinks or consumers (u USI).
The capital resource constraint to invest in capacity expansion is the sum of the investment cost
of the units given by fixed and varying terms as shown in Eq. (6). The discrete decision to invest
in the unit u, the binary variable yu,t, represents the fixed term considering the parameter βu,t as
fixed cost in $. The variable term is related to the size of the unit capacity expansion QEu,t with
parameter αu,t as cost per unit of volume/mass added in the capacity expansion. Eqs. (1)-(2) and
(6) represent the design level formulation. Eqs. (4) and (5) are the operational level constraints.
Eq. (3) is the linking constraint between the levels. The CCIP problem maximizes the net present
value (NPV) of the profit (revenue minus feed costs and capital costs) over time as seen in Eq.
(7), where prs,t and cu,t are stream prices and operational costs, respectively. These coefficients
have simple discounts in the NPV objective function according to an inflation or deflation rate
parameter, which can also be time-varying or time-dependent. The time-horizon is usually
modeled over several years, where the time-periods are in years or sub-years such as quarters.
These time-periods can also be non-uniform in the sense that their durations can be variable, but
exogenously defined (known a priori). These types of problems may have different aggregated or
disaggregated formulations such as lot-sizing reformulations (Liu and Sahinidis, 1996) which are
not applied explicitly in this model.
MILP planning models based on NPV maximization for investments in chemical plants solved
the CCIP problem using a combination of integer cuts and branch and bound (Sahinidis,
Grossmann, Fornari, and Chathrathi, 1989), polyhedral projection and strong cutting planes (Liu
and Sahinidis, 1996), both considering only the design model, and bi-level decomposition

9. 9
approach (Iyer and Grosmann, 1998), where binary decisions for design (capacity expansion or
installation) and operation (setup or shutdown) of a process network are made separately. For
large problems, solutions with 10% higher NPV are obtained in the decomposed method when
compared to the suboptimal solution of the full space model.
Van den Heever and Grossmann (1999) propose both outer-approximation (Duran and
Grossmann, 1986) and bi-level decomposition (Iyer and Grosmann, 1998) strategies for the
design and operational planning of simple process industry networks, incorporating design,
operational, and capacity planning for design and operational cost minimization models. They
use disjunctive programming techniques (Raman and Grossmann, 1994) to extend the
methodology to the case of multi-period design and planning of nonlinear chemical process
systems. Their work addresses the problem of the computational expense in solving the MILP
step, which is often the bottleneck in the computations of multi-period optimization problems in
MINLP’s that involve discrete decisions for topology selection, capacity expansion, and
operation at each time period.
Jackson and Grossmann (2002) propose a high-level MILP model to address the retrofit design
of process networks using an economic objective function to allow multiple types of
modifications of capacity and capability improvements considering them as simple coefficients
based on the discrete decision of the project approval. Capacity is related to process unit
throughputs increase and capability to chemical conversion such as catalyst bed activity in
cracking, reforming and hydrotreating units. When compared to big-M constraints, the examples
illustrate the robustness of the generalized disjunctive programming approach with convex hull
formulation (Balas, 1985), which gives a tight LP relaxation and leads to faster solution times.

10. 10
For CCIP problems including supply chain management, You and Grossmann (2008) use a
quantitative approach for designing responsive supply chains under demand uncertainty, in
which strategic, tactical, and operational decisions (e.g. installation of plants, selection of
suppliers, manufacturing sites, distribution centers, and transportation links) are integrated with
the scheduling decisions (e.g. product transitions and changeovers) for the multi-site and multi-
echelon process supply chain network. The expected lead time was proposed as a measure of
process supply chain responsiveness. A multi-period MINLP model was developed for the bi-
criterion optimization of economics and responsiveness, while considering customer demand
uncertainty. Multi-period and multi-site planning models predicting plant’s optimal capacity,
production levels and sale profiles inside supply chains (You, Grossmann and Wassick, 2011;
Corsano et al., 2014) showed that the bi-level decomposition (Iyer and Grossmann, 1998)
requires smaller computational times leading to solutions that are much closer to the global
optimum when compared to the full space solution and to Lagrangean decomposition (Guignard
and Kim, 1987).
Within the oil-refining industry, design and coordination of multi-site facility network considers
capacity expansion of existing process units using MILP with an overall objective of minimizing
total annualized cost (Al-Qahtani and Elkamel, 2008; 2009). Recently, Menezes et al. (2014a,
2014b) addressed approaches with aggregation in process unit capacity considering all current
and future refineries in Brazil in one hypothetical large refinery to approximate the current and
future fuel production and import scenarios in the country. Menezes et al. (2014a) proposed a
multi-period process design MINLP model for predicting overall capacity expansion of existing
oil-refinery units considering scenarios for different future fuel markets. The authors also
proposed a scenario-based approach where a single-period NLP model in an operational planning

11. 11
fashion simulates future process design (with respect to market scenarios) to avoid mixed-integer
models (Menezes et al., 2014b). In this case, unit throughput upper bounds of the existing units
in the hypothetical large refinery are considered a large number enough to fulfill market
demands. The differences between unit throughputs in the current and future market scenarios
are the required capacity expansions, so that this NLP approach disregards both binary variables
for project approving and variables of unit capacity (QCu,t) and expansions (QEu,t) as seen in the
CCIP formulation in Eqs. (1)-(7). Both aggregated approaches yield better results in terms of
fulfilling market demands with lower capital investment and higher NPV, when compared with
national plans on unit capacity additions proposed for the new refinery sites. For the national
strategic planning level, the aggregated model is satisfactory for predicting overall capacity
expansion per type of oil-refinery unit needed to match future fuel demands in the country, and
to prevent the solution of very large models that includes all the refineries.
Menezes, Kelly and Grossmann (2015) proposed a phenomenological decomposition heuristic
(PDH) by solving the CCIP problem for integrated multi-refineries with intermediate stream
transfers between them. The work handles oil-refinery optimization including mixing of crude-
oils, processing units, and blending of product streams for expansion of existing units and
installation of new units considering time of execution for the projects as the duration of the
intervals, so that when the project is approved its full implementation is accounted in the
following time intervals. The PDH is a decomposition formulation solving MILP (quantity-logic)
and NLP (quantity-quality) sub-problems iteratively, where nonlinearities from quantity and
quality balances are linearized or neglected in the MILP master problem, and the found design
solution (project schedules and their sizes, etc.) are fixed in the NLP sub-problem where yields

12. 12
and rates are updated and sent back to the MILP problem until the MILP and NLP sub-problems
convergence on key variables.
Two recent papers reviewed strategic planning problems in the process industry. Martínez-Costa,
Mas-Machuca, Benedito, and Corominas (2014) describe and analyze strategic capacity planning
problems and their proposed mathematical programming modeling in manufacturing. In Sahebi,
Nickel and Ashayeri (2014) a very detailed taxonomy lists strategic and tactical planning types
of problems within the crude oil supply chain (COSC). They survey fifty-four (54) papers related
to COSC planning problems between 1988 and 2013, considering oil reserve and production
until fuel deliveries to clients. Ongoing and emerging challenges surrounding strategic and
tactical decisions of COSC problems are investigated and gaps in the literature are analyzed to
recommend possible research directions. The 54 papers reviewed in the COSC field do not
address the issues introduced here as installations of units, project staging, regards to the
different yields as well as project execution, sequence-dependency between stages, etc.
Further to these approaches, is the novel model presented in this paper in which the capital
investment planning problem is reformulated using sequence-dependent setups (Kelly and
Zyngier, 2007) to include stages of project execution. In addition, capital and capacity are
regarded as flows or amounts in a scheduling context. We include expansion and installation of
units or equipment modeled in a non-aggregated framework, i.e., in an actual or real plant model.
In this case, considering a multi-period formulation, the model gives rise to large-scale MINLP
problems in which the input-output approximations using sequence-dependent setups modeling
is proposed for solving industrial-sized problems in a MILP model instead.
3. Sequence-Dependent Setup Modeling of Stages

13. 13
At this point we describe the sequence-dependent setups, changeover, or switchover discrete-
time modeling (Kelly and Zyngier, 2007) to optimize the selection of projects considering their
stages in the capital investment planning problem. Unlike conventional approaches such as the
full space (Sahinidis, Grossmann, Fornari, and Chathrathi, 1989; Liu and Sahinidis, 1996;
Menezes et al., 2014a), the bi-level decomposition (Iyer and Grosmann, 1998; You, Grossmann,
and Wassick, 2011; Corsano et al., 2014), the generalized disjunctive programming formulation
(Van den Heever and Grossmann, 1999; Jackson and Grossmann, 2002), and the
phenomenological decomposition heuristic (Menezes, Kelly, and Grosmann, 2015), we address
explicitly project setup and its stages over time by modeling it essentially as a scheduling
problem. When complex process frameworks are modeled, like those found in the oil-refining
industry, project lifetime must be included to better assess capital resource predictions and
production discounts in the NPV function. In this sense, the conventional capital investment
planning approaches are more suitable for repair or retrofit problems, where disregarding project
execution timetable and related changes in the production has little influence in the decisions
because of the lower capital investment involved and lower project impacts in the production. In
the end, any kind of improvement in the representation of project scheduling and staging within
the oil-refining industry can potentially save millions, if not billions of U.S. dollars, from the
shorter term repair types of projects to the longer term revamps and installations of process units.
3.1. Types of Capital Investment Planning
Considering shorter to longer term projects with stages in a capital investment planning (CIP),
the types of projects can be classified as revamping (facilities planning), retrofitting
(capacity/capability planning), and repairing (maintenance/turnaround planning). We denote the
proposed model as the generalized capital investment planning (GCIP) problem extending the

14. 14
conventional capital investment planning (CCIP), and specifically for the NPV-based capacity
planning of existing units or retrofit problem as discussed in Sahinidis, Grossmann, Fornari, and
Chathrathi (1989) and Liu and Sahinidis (1996). CCIP is the optimization problem where it is
desired to expand the capacity and/or extend the capability (conversion) of either the expansion
of an existing unit (Jackson and Grossmann, 2002) or the installation of a new unit (Menezes,
Kelly, and Grosmann, 2015) without considering impacts of inherent stages of projects.
Fig. 1 shows the three types of CIP problems with its typical capital investment cost and time
scales.
Fig. 1. Three types of capital investment planning problems.
The short-term CIP problem is the repair problem, which is typically referred to as maintenance
planning or turnaround and inspection (T & I) planning, and has a correction stage that is placed
in between the existing unit before the correction and the improved unit after the correction. For
repair problems, the correction stage is in-series, and is the stage that implements the turnaround

15. 15
and inspection activities such as de-fouling or cleaning heat exchangers, etc. In this case, the
existing process can be totally or partially shutdown during the corrections.
The medium term CIP problem is the retrofit problem (i.e., replacing or refitting new or
enhanced equipment after it has already been constructed and in production), and is often
referred to as capacity planning or production design synthesis, and has a commission stage that
is placed in between the existing unit before the commission and the expanded/extended unit
after the commission. For retrofit problems, there can be a construction stage in-parallel to the
existing stage and the commission stage is in-series similar to a cleaning/purging (or repetitive-
maintenance) operation, activity, or task in sequence-dependent setup, changeover, or switchover
problems. The existing process can be partially or totally shutdown during the commission stage.
The long-term CIP problem is the revamp problem, which is sometimes referred to as facilities
planning and process design synthesis, and has a construction stage that is placed in between the
existing unit before the construction and the expanded/extended unit after the construction. If the
unit does not previously exist, then this is an installation versus an expansion/extension. For
revamp problems, the construction stage is in-series, and is the stage that installs the expanded or
extended equipment. The existing process is totally shutdown during the construction or revamp
stage. For an installed new unit, a commission phase can be placed between the construction and
installed stages, in which, as an example, parts of the new process unit are started up before its
full implementation. To accommodate multiple stages within the same planning horizon,
multiple pre-preparation, commission and expansion operations need to be configured where
sequence-dependent setups or switchovers can only occur from a previously expanded operation
through a commission operation to the next expanded operation as a phasing sequence.

16. 16
To the best of our knowledge, this is the first time a connection between the different types of
CIP problems has been made, i.e., repair/correction (operational), retrofit/commission
(tactical/debottlenecking) and revamp/construction (strategic). There are two other salient
aspects of our general CIP formulation that sets our formulation apart from all other formulations
found in the literature. The first is the modeling of sequence-dependent setups, switchovers, or
changeovers to manage the realistic situation that a correction, commission, or construction
stage, activity or task must be planned or scheduled in between the existing and
expanded/extended units. This is handled using the appropriate variables and constraints found in
Kelly and Zyngier (2007), which albeit intended for discrete-time scheduling problems with
repetitive-maintenance, can be easily applied to CIP problems that are also modeled in discrete-
time given the longer term decision-making framework. The second aspect is the modeling of
capacity/capability and capital as flows or quantities. This is the notion that the correction,
commission and construction stages actually produce or create capacity and/or capability, which
can then be used or consumed by the unit in subsequent time-periods, yet there is of course a
charge for the capacity/capability known as the capital cost expenditure.
3.2. Sequence-Dependent Setup Formulation
The sequence-dependent formulation addressed in Kelly and Zyngier (2007), embedded in
IMPL’s semantic modeling, considers transitioning variables as continuous variables that rely on
the independent binary variable unit-operation-time setup yu,m,t ofa unit u in mode m at time t.
This setup variable can be configured for initial, intermediate, or final states of a unit with
regards to the use of modes to manage the project staging with temporal details of the unit-
operation (u,m) where its transitioning setup variable of an operation suu,m,t, its shutdown sdu,m,t,
and switchover-to-itself swu,m,m,t are relaxed within the interval [0,1], so we do not need to

17. 17
explicitly declare them as binary variables in the branch-and-bound search of the MILP. Hence,
they will not be part of the search-tree because their integrality is implied by the tight constraints
Eqs. (8)-(10) and reduces the size of the search-tree where Fig. 2 illustrates their values for a
batch-process with fixed batch-size and variable batch-time (FSVT) and is similar to continuous-
process temporal transitioning.
𝑦 𝑢,𝑚,𝑡 − 𝑦 𝑢 ,𝑚,𝑡−1 − 𝑠𝑢 𝑢,𝑚,𝑡 + 𝑠𝑑 𝑢 ,𝑚,𝑡 = 0 ∀ 𝑢, 𝑚, 𝑡 (8)
𝑦 𝑢,𝑚,𝑡 + 𝑦 𝑢 ,𝑚,𝑡−1 − 𝑠𝑢 𝑢,𝑚,𝑡 − 𝑠𝑑 𝑢 ,𝑚,𝑡 − 2𝑠𝑤 𝑢,𝑚,𝑚,𝑡 = 0 ∀ 𝑢, 𝑚, 𝑡 (9)
𝑠𝑢 𝑢,𝑚,𝑡 + 𝑠𝑑 𝑢,𝑚,𝑡 + 𝑠𝑤 𝑢 ,𝑚,𝑚,𝑡 ≤ 1 ∀ 𝑢, 𝑚, 𝑡 (10)
In Eq. (8), the setup of a unit-operation-time binary variable yu,m,t is selected between its startup
suu,m,t and its shutdown sdu,m,t, because the past unit-operation-time binary variable yu,m,t-1
guarantees if the setup variable is still selected within the next time period (when the startup
returns to zero), its shutdown is zero (sdu,m,2 = 0). Instead, if the setup is zero (yu,m,t = 0), its
shutdown should be one (sdu,m,2 = 1). Eq. (9) introduces the switchover-to-itself swu,m,m,t
continuous variable meaning that if the unit-operation-time setup variable is still true (the past
and current setups are selected then), the swu,m,m,t = 1 and the startup and shutdown should be
zero. Eq. (10) preserves the integrality of the continuous variables suu,m,t and sdu,m,t preventing
the solution from setting both suu,m,t and sdu,m,t to 0.5 in the LP nodes of the MILP branch-and-
bound search.

18. 18
Fig. 2. Scheduling stages for a batch process taken from Kelly and Zyngier (2007).
To model the sequence-dependent transitioning of project stages in the proposed capital
investment planning problem, we use the concept of “memory” variables first described in Kelly
and Zyngier (2007). This is also a continuous variable that tracks the temporal unit-operation
(u,m) events or activities occurring for each unit within the time horizon and the memory yyu,m,t
of the last operation performed, thus allowing us to know the last production operation or state
that was active for the unit. Equations (11)-(14) demonstrate the relations of the memory variable
in the sequence-dependent formulation.
∑ 𝑦𝑦 𝑢,𝑚,𝑡 = 1
𝑚
∀ 𝑢, 𝑡 (11)
𝑦 𝑢,𝑚,𝑡 − 𝑦𝑦 𝑢,𝑚,𝑡 ≤ 0 ∀ 𝑢, 𝑚, 𝑡 (12)
𝑦 𝑢,𝑚,𝑡 − 𝑦𝑦 𝑢,𝑚,𝑡−1 − 𝑠𝑢 𝑢,𝑚,𝑡 ≤ 1 ∀ 𝑢, 𝑚, 𝑡 (13)

19. 19
𝑦 𝑢,𝑚,𝑡−1 − 𝑦𝑦 𝑢,𝑚,𝑡 − 𝑠𝑑 𝑢 ,𝑚,𝑡 ≤ 1 ∀ 𝑢, 𝑚, 𝑡 (14)
By Eq. (11), if even any operation m is not being performed in the unit u (yu,m,t = 0), the
information on the past productive operation is preserved by the memory variables yyu,m,t. This is
a single-use or unary-resource or commitment constraint that states that one and only one
production operation must be active or setup on the unit in any given time-period. In Eq. (12),
when a unit is performing a particular production operation, the appropriate memory variables
yyu,m,t are activated. Eqs. (13)-(14) propagate the memory of the productive operation when the
unit is completely shutdown or inactive during the productive to the non-productive transitions
and vice-and-versa. Eq. (13) is when the unit goes from the non-productive to the productive
state, so it is a startup suu,m,t. Eq. (14) is when the unit goes from the productive to the non-
productive state, so it is a shutdown sdu,m,t. Eqs. (11)-(14) are applied even for units that always
have an operation active throughout the horizon such as a storage unit (tank), given that
sequence-dependent switchovers from one operation to another must be properly tracked; in this
case, the unit-operation is in a switchover-to-itself, sw u,m,m,t. Fig. 2 shows the stages of two
batches and the profiles of the independent variable (yu,m,t) and the four dependent variables
(suu,m,t, sdu,m,t, swu,m,m,t, and yyu,m,t) extracted from Kelly and Zyngier (2007).
In order to activate a specific maintenance, non-productive, intermediate operation on a unit
before a spatial switchover occurs from operation m in some time-period in the past to m’, then
Eq. (15) must be used, where m is the from/previous operation or operation-group; m’ is the
to/next operation or operation-group and r is the repetitive-maintenance-intermediate, which
represents in the generalized capital investment planning (GCIP) problem the project stage
(correction, commission, and construction). In this equation we expect that the startup of m’ and
the shutdown of m take place in the same time-period i.e., they are temporally coincident.

20. 20
(𝑦𝑦 𝑢,𝑚′,𝑡 + 𝑦𝑦 𝑢,𝑚,𝑡−1 − 1) + 𝑠𝑢 𝑢,𝑚′,𝑡 − 𝑠𝑑 𝑢,𝑟,𝑡 ≤ 1 ∀ 𝑢, 𝑚, 𝑚′
, 𝑟, 𝑡 (15)
This equation allows us to know when the existing or non-existing operation occurs on the unit,
and then we can insert the commission or construction stage for example before the expanded or
installed operations or stages; this is the core idea in GCIP with the explicit commission and
construction stages activated in between the existing/non-existing and the expanded/installed
operations or stages.
The sequence-dependent setup, changeover, or switchover relationship between different
operations i.e., “phasing”, “purging”, “prohibiting” and “postponing” on the same unit can be
derived from these dependent variables, whereby intermediate operations can be activated and
placed in between the mode operations such as the project execution phases proposed in this
work, i.e., the correction, commission or construction stages. The mode or operation setup
variable yu,m,t, in the proposed GCIP model are defined as (“Existing”, ”Non-Existing”) and
(“Expanded”, ”Extended”, ”Installed”) stages of a capacity investment planning problem in
which the input-output yields, rates, etc. for each operation on the same unit can have different
values. Increasing the number of project stages between the initial and final project state can
improve the accuracy of the problem.
We introduce staging or phasing as variation of the sequence-dependent changeover problem
(Kelly and Zyngier, 2007, Balas et. al., 2008), except that the sequencing, cycling, or phasing is
assumed to be fixed as opposed to being variable or free. Phasing allows for the implementation
of what is known in the specialty chemicals and consumer goods industries as a product-wheel,
and also known as blocking in other industries where the cost of sequence-dependent
changeovers is significant such as in the paper and bottling industries. A product-wheel forces

21. 21
product A to be followed by product B then followed by product C and so on with a rigid
sequencing. In this way, the sequence-dependency is fixed or forced, i.e., it is essentially pre-
defined, as opposed to variable or free sequence-dependent switchovers, requiring more
variables and constraints to be modeled, and more CPU time when solving or searching for
solutions. Hence, the advantage of phasing is that it can be used to find solutions quicker at the
expense of being less flexible in terms of handling more disruptions or disturbances with respect
to supply, demand, investment, maintenance, and other production-order scenarios.
The other three sequence-dependent changeover modeling types are what we call purging,
prohibiting, and postponing. Purging requires a repetitive-maintenance task to be configured
between two production or process operations involving cleaning activities that may or may not
require the consumption and/or production of resources, which in the capital investment planning
case, can be considered the configuration or re-configuration task (correction, commission, and
construction). Prohibiting disallows certain sequences of operations from ever being scheduled
or occurring like as in a multi-product pipeline or blender in which certain sequence of products
are strictly not allowed or forbidden to avoid or decrease product contamination. Postponing
implements sequence-dependent and sequence-independent down-times between certain
operations, modes or stages of a unit.
Other project scheduling types of problems in both discrete and continuous time can be found in
Kopanos et al. (2014) to address the resource-constrained project scheduling problem (RCPSP)
in which renewable resources fully retrieve the occupied resource amount after the completion of
each activity, while the total duration of the project (i.e., the makespan) is minimized satisfying
precedence and resource constraints. The problem consists of finding a schedule of minimal
duration by assigning a start time to each activity in which the precedence relations and the

22. 22
resource availabilities are respected. Several planning and process level decision problems can be
reduced to the RCPSP (Varma et al., 2004) such as in high scale projects management in
software development, plant building, and military industry (Pinedo and Chao, 1999), and in
highly regulated industries where a large number of possible new products are subject to a series
of tests for certification (Shah, 2004) such as in pharmaceutical and agrochemical industries.
4. Generalized Capital Investment Planning (GCIP) Model
The network of the generalized approach for capital investment planning is represented in the
flowsheet-based superstructure shown in Fig. 3, where capacity and capital are treated as flows
in a scheduling environment. The same idea could be used for capability (as conversion) as well,
although in this case a simple coefficient is able to represent the ratio of conversion. The
generalized CIP formulation can be applied to any CIP problem found in the process industries,
which may require more scheduling details to be considered. It can be easily modeled using the
modeling and solving platform IMPL that is based on the unit-operation-port-state superstructure
(UOPSS) formulation as seen in the following sub-section. This new modeling platform allows
the modeler or user the ability to configure the problem using semantic variables such as flows,
holdups, yields and setups, startups, etc. that is more intuitive and natural without having to
explicitly code the sets, parameters, variables and constraints required in algebraic modeling
languages such as AIMMS, AMPL, GAMS, LINGO, MPL, MOSEL, OPL, etc.

23. 23
Fig. 3. Motivating example 1: small GCIP flowsheet for expansion.
In the GCIP model shown in Fig. 3, the diamond shapes ( ) or objects are the perimeter unit-
operations where they consume material A (source) and produce material B (sink), i.e., in and
out-bound resources. We have just one process unit with three operations of Existing,
Commission, and Expanded as shown by the square boxes with an “x” (cross-hairs) through it
indicating it is a continuous-process type ( ). The first dotted line box highlights that only one
unit-operation can be active or setup at a time, i.e. a unary resource or the unit commitment
constraint. The small circles are the in-ports ( ) and out-ports ( ) where these ports have the
attributed lower and upper yields available similar to the modeling of generalized network-flow
problems, i.e. having the previously mentioned Leontief input-output models with intensities,
bill-of-materials, or transfer coefficients. The port-states allow flow into and out of a unit and
can be considered as flow-interfaces similar to ports on a computer, i.e., nozzles, spouts, spigots.
Port-states also provide an unambiguous description of the flowsheet or superstructure in terms
of specifically what type of materials or resources are being consumed and produced by the unit-

24. 24
operation. Port-states can also represent utilities (steam, power), utensils (operators, tools) as
well as signals such as data, time, tasks, etc.
Each of the two perimeters, A and B, can have tanks ( ) available for storage, and is a
requirement when balancing the production-side and transportation-side supply and demand of
the value-chain for example. Finally, the lines or arcs ( ) between the unit-operations and port-
states and across an upstream unit-operation-port-state to a downstream unit-operation-port-state
correspond to flows (external streams) as one would except given that the superstructure is
ultimately composed of a network, graph or diagram of nodes/vertices and arcs/edges (directed).
Each unit-operation and external stream have both a quantity and a logic variable assigned or
available, and represent either a flow or holdup if quantity and either a setup or startup if logic.
Batch-processes have holdups and startups, continuous-processes have flows and setups, pools
have holdups and setups, and perimeters only have a logic setup variable. The internal streams
(lines with no arrow-head) have neither explicit/independent flow and setup variables given that
their flows are uniquely determined by the aggregation of the appropriate external streams, and
their setups are taken from the setup variables on the unit-operation they are attached to. The
network material balance is given by the flowsheet connectivity between the elements or shapes
(units, tank, in-ports, out-ports, etc.).
The case in Fig. 3 shows the expansion of an existing process unit (Process), which is initially in
the Existing mode. For a completely grass-roots or green-field installation, the operations or
states are the Construction and the Installed modes, which are equivalent to the Commission and
Expanded modes depicted in Fig 3. However, to control the sequence of stages, a NonExisting
mode should be included for installations that is equivalent or symmetrical to the Existing mode
for the expansions. Each of the expansion or installation unit-operations represented by the

25. 25
Commission/Construction stage, have a capacity port-state cpt connected to the unit-operation
named Charge to transfer this capacity to the Capacity tank where the Expanded or Installed unit
can have an increase in capacity by the new additional charge-size from the tank. In the Charge
unit, the capital port-state cpl carries the NPV cash-flow to the perimeter named Capital
(diamond shaped). This is the non-material or non-stock flow of a financial resource.
The Existing or Non-Existing unit-operation selection is based on the economic viability with
respect to its expected Expanded or Installed cost and projected revenue of the products, which
in the example in Fig. 3 is the perimeter B. The inlet port-state of this unit-operation will have a
time-varying NPV cash-flow lower, and upper bound to constrain the expansions and/or
installations according to the expected cash-flow profiles in the future. An additional restriction
required, sometimes referred to as a side-constraint, is the fact that if an expansion/installation
unit-operation is selected in some future time-period, then it must be setup for the rest of the
time-horizon. This can be modeled using the uptime logic constraint, where a lower or minimum
uptime is configured as the time-horizon length of the problem (Wolsey, 1998, Kelly and
Zyngier, 2007 and Zyngier and Kelly, 2009). Uptime is also known as a run- or campaign-
length, and essentially restricts a shutdown of the unit-operation for a specified number of time-
periods in the future.
4.1. Unit-Operation-Port-State Superstructure (UOPSS) Formulation
Fig. 4 shows the UOPSS scheme described in Kelly (2004), Kelly (2005) and Zyngier and Kelly
(2012), where unit-operations ( ) and streams/links ( ) between in-ports ( ) and out-ports (
) have a binary variable to turn on or off the shapes or structures (units-operation, ports, and
streams) over time. This idea permits to solve industrial-sized optimization problems in both

26. 26
planning and scheduling environments in which the UOPSS structures and procedures are
integrated over space and time, considering renewable (units) and/or non-renewable resources
(states).
Fig.4. UOPSS scheme.
The IMPL built-in technique to construct equations based on the network connectivity using the
UOPSS scheme (constraint generation) considers as main variables flow f and setup su of unit-
operations and streams as seen in Fig. 4. Flows are continuous variables represented as x in
v2r_xmfm,t and v3r_xjifj,i,t and setups are binary variables represented as y in v2r_ymsum,t and
v3r_yjisuj,i,t. In IMPL’s constraint generation, lower and upper bound constraints are built
considering the configured lower and upper bounds of both continuous and binary variables. As
IMPL uses numbers instead of names to construct the problem faster and using less memory, the
unit-operation setup can be reduced only to the operation m, as each m for each unit u has a
unique number to represent its setup in the formulation (see the shape-number check in the
supplementary material, page 4). So, the overall formulation is shown in equations (16)-(29).
𝑣2𝑟_𝑥𝑚𝑓𝑚,𝑡 ≥ 𝐿𝐵𝑓 𝑚 𝑣2𝑟_𝑦𝑚𝑠𝑢 𝑚,𝑡 ∀ 𝑚, 𝑡 (16)

27. 27
𝑣2𝑟_𝑥𝑚𝑓𝑚,𝑡 ≤ 𝑈𝐵𝑓𝑚 𝑣2𝑟_𝑦𝑚𝑠𝑢 𝑚,𝑡 ∀ 𝑚, 𝑡 (17)
∑ 𝑣3𝑟_𝑥𝑗𝑖𝑓𝑗,𝑖,𝑡
𝑗∈(𝑗,𝑖)
≥ 𝐿𝐵𝑓 𝑚 𝑣2𝑟_𝑦𝑚𝑠𝑢 𝑚,𝑡 ∀ (𝑖, 𝑚), 𝑡
(18)
∑ 𝑣3𝑟_𝑥𝑗𝑖𝑓𝑗,𝑖,𝑡
𝑗∈(𝑗,𝑖)
≤ 𝑈𝐵𝑓𝑚 𝑣2𝑟_𝑦𝑚𝑠𝑢 𝑚,𝑡 ∀ (𝑖, 𝑚), 𝑡
(19)
∑ 𝑣3𝑟_𝑥𝑗𝑖𝑓𝑗,𝑖,𝑡
𝑖∈(𝑗,𝑖)
≥ 𝐿𝐵𝑓 𝑚 𝑣2𝑟_𝑦𝑚𝑠𝑢 𝑚,𝑡 ∀ (𝑚, 𝑗), 𝑡
(20)
∑ 𝑣3𝑟_𝑥𝑗𝑖𝑓𝑗,𝑖,𝑡
𝑖∈(𝑗,𝑖)
≤ 𝑈𝐵𝑓𝑚 𝑣2𝑟_𝑦𝑚𝑠𝑢 𝑚,𝑡 ∀ (𝑚, 𝑗), 𝑡
(21)
𝑣3𝑟_𝑥𝑗𝑖𝑓𝑗,𝑖,𝑡 ≥ 𝐿𝐵𝑓𝑗,𝑖 𝑣3𝑟_𝑦𝑗𝑖𝑠𝑢 𝑗,𝑖,𝑡 ∀ (𝑗, 𝑖), 𝑡 (22)
𝑣3𝑟_𝑥𝑗𝑖𝑓𝑗,𝑖,𝑡 ≤ 𝑈𝐵𝑓𝑗,𝑖 𝑣3𝑟_𝑦𝑗𝑖𝑠𝑢 𝑗,𝑖,𝑡 ∀ (𝑗, 𝑖), 𝑡 (23)
∑ 𝑣3𝑟_𝑥𝑗𝑖𝑓𝑗,𝑖,𝑡
𝑗∈(𝑗,𝑖)
≥ 𝐿𝐵𝑦𝑖,𝑚 𝑣2𝑟_𝑥𝑚𝑓𝑚,𝑡 ∀ (𝑖, 𝑚), 𝑡
(24)
∑ 𝑣3𝑟_𝑥𝑗𝑖𝑓𝑗,𝑖,𝑡
𝑗∈(𝑗,𝑖)
≤ 𝑈𝐵𝑦𝑖,𝑚 𝑣2𝑟_𝑥𝑚𝑓𝑚,𝑡 ∀ (𝑖, 𝑚), 𝑡
(25)
∑ 𝑣3𝑟_𝑥𝑗𝑖𝑓𝑗,𝑖,𝑡
𝑖∈(𝑗,𝑖)
≥ 𝐿𝐵𝑦 𝑚,𝑗 𝑣2𝑟_𝑥𝑚𝑓𝑚,𝑡 ∀ (𝑗, 𝑚), 𝑡 (26)
∑ 𝑣3𝑟_𝑥𝑗𝑖𝑓𝑗,𝑖,𝑡
𝑖∈(𝑗,𝑖)
≤ 𝑈𝐵𝑦 𝑚,𝑗 𝑣2𝑟_𝑥𝑚𝑓𝑚,𝑡 ∀ (𝑗, 𝑚), 𝑡
(27)
∑ 𝑣2𝑟_𝑦𝑚𝑠𝑢 𝑚,𝑡
𝑚(𝑚∈𝑢)
≤ 1 ∀ 𝑢, 𝑡
(28)

28. 28
𝑣2𝑟_𝑦𝑚𝑠𝑢 𝑚′
,𝑡 + 𝑣2𝑟_𝑦𝑚𝑠𝑢 𝑚,𝑡 ≥ 2 𝑣3𝑟𝑦𝑗𝑖𝑠𝑢 𝑗 ,𝑖,𝑡
∀ ( 𝑚′
, 𝑗), (𝑖, 𝑚) (29)
Eqs. (16)-(21) are related to the setup variable of the unit-operation m represented by
v2r_ymsum,t. Eqs. (16) and (17) means that if the unit-operation m is active by its setup, the unit-
operation flow v2r_xmfm,t should be between the lower (LBfm) and upper (UBfm) bounds of the
flows also known as semi-continuous constraints. It is similar to minimum and maximum
throughputs of process units or expansion/installation sizes. Eqs. (18)-(21) interconnect unit-
operations to ports so the flow of the streams arriving or leaving unit-operations are also between
their bounds. Eqs. (22) and (23) considers stream setups v3r_xjisuj,i,t to turn-on or turn-off the
lower (LBfj,i) and upper (LBfj,i) bounds of the streams v3r_xjifj,i,t flowing from out-ports j to in-
port i of unit-operations m forming the subset j,i determined by the UOPSS connectivity
network. In the supplementary material lines 24-66, the section “Construction Data (Pointers)”
presents the list of connections for motivating example 1 in Fig. 3.
Eqs. (24) and (25) consider bounds on yields since unit-operations m can have more than one
feed stream given by i,m subsets and each one of the in-ports i can still have a number of
interconnected out-ports j giving by j,i subsets. It is the same for the products leaving the unit-
operations as in Eqs. (26) and (27). In Eq. (28) the unit-operations procedures, modes or tasks
permitted in the physical unit u is at most one. Eq. (29) is the structural transition constraint
similar to the sequence-dependency constraints found in William (1999) which is a core idea of
the UOPSS scheme to manage the up to downstream flows. It says that only if the setup of unit-
operations of different units interconnected by streams within out-ports j and in-ports i are
turned-on, the setup of the stream v3r_xjisuj,i,t may be turned-on if required. If one unit-operation
is turned-off, the stream setup is off by implication. The remaining constraints of the GCIP

29. 29
model using sequence-dependent setups will be given using the motivating example 1 in the
following.
4.2. Motivating Example 1
We explore further the solution to the small retrofit generalized CIP problem fully defined in Fig.
3. In the supplementary material example’s IMPL configuration (lines 1 to 151) and formulation
(lines 153 to 487) is given. In the motivating example 1 the future planning time-horizon is
arbitrarily configured as three months and with one month time-periods. The existing or old
capacity of the process is 1.0 quantity-units per month and the new capacity can be 1.5. The
capital cost is computed with α = 0.5 ($ per quantity-units) and β = 0.5 ($ per setup-units). All of
these data are declared in the “Calculation Data (Parameters)” section in the supplementary
material (lines 4 to 17). The network in Fig. 3 using the UOPSS configuration creates the
“Construction Data (Pointers)” section (lines 24 to 66). By simplification, the cost for material A
is $0.0 per quantity-unit and the price for B is $1.0 per quantity-unit, and we do not apply any
NPV given the relatively short horizon. The costs are defined in the “Cost Data (Pricing)”
section in the supplementary material (lines 121 to 128).
The operations of Commission and Expanded each have a special out-port and in-port labeled as
cpt, which stands for the outflow and inflow of capacity, respectively. There is a Charge unit-
operation, which will only be setup if the Commission unit-operation is active, and its purpose is
to convert the variable capacity to a variable and fixed capital cost. It is represented as
“Charge,,cpl,,alpha,alpha,beta-alpha*oldcapacity” in the “Capacity Data (Prototypes)” section
(see supplementary material, line 97), where the α (alpha) coefficient or parameter is applied to
the incremental or delta capacity change, and the β (beta) is applied to the setup variable if the

30. 30
Charge unit-operation is on or open. The out-port on the Charge unit-operation labeled cpt is the
flow of capacity charged or dispatched to the Capacity pool unit-operation (triangle shape).
During the one time-period when the Charge unit-operation is active, the flow of capacity to the
Capacity pool must have enough capacity to operate the Expand operation for as many time-
periods left in the planning time-horizon. For example, if we have a three time-period future
horizon and the Commission operation starts in time-period one, then enough capacity must be
fed or sent to the Capacity pool unit-operation for time-periods two and three (END). In this
way, the capacity to be expanded or installed, an extensive amount, proxies as an intensive value.
As example, for the first period, it is represented as “Charge,,cpt,,1.0*(END-1.0),1.0*(END-
1.0),,0.0,1.0” in the “Command Data (Future Provisos)” section (see supplementary material,
lines 138 to 151).
The cpt in-port on the Expanded operation will draw only up to the maximum allowable or upper
limit of the expanded capacity allowed from the Capacity pool, and this will control the capacity
charge-size, throughput or flow through the unit-operation for the Expanded operation. In order
to do this, the lower yield bound on the cpt in-port is configured as one and the upper yield
bound as infinity or some large number. This will regulate the capacity of the Expanded unit-
operation as in “Process,Expanded,cpt,,1.0,large” in the “Capacity Data (Prototypes)” section
(see supplementary material, lines 67 to 100).
The problem is solved using MILP with an optimal objective function of $3.25. If we apply no
expansion, then the profit would be $3.0 since the existing capacity is 1.0 for three time-periods.
Since the profit is $0.25 more than $3.0, then there has been an expansion where the
Commission operation is setup or started in time-period one. To perform an expansion of 1.5 –
1.0 = 0.5 quantity-units then, the capital cost required is 0.5 * 0.5 + 0.5 = $0.75. Given the timing

31. 31
of the Commission stage, this implies that the Expanded stage occurs in time-periods two and
three, which is enforced by the sequence-dependent setup modeling, i.e., after the Commission
stage only the Expanded stage can be setup for the rest of the horizon. With an expansion capital
cost of $0.75 and a revenue for the sale of material B of 1.0 + 1.5 + 1.5 = $4.0 for the three
planning periods, this yields a profit of $4.0 - $0.75 = $3.25, which is the same value found by
the MILP. The Gantt chart for this example is found in Fig. 5. In this case, we are considering
the commission mode with the same capacity as in the existing to permit production during this
stage, since an interruption of this production in the first time period would impede the
expansion. The required data to control the expansion is defined in the three frames from line
101 to 120. In the “Constriction Data (Practices/Policies)” section the uptime for the
Commission (with duration of one time-period since the lower and upper uptime are set to 1) and
for the Expansion with maximum uptime set to END that is equal three time-period. In the
“Consolidation Data (Partitioning)” section, the initial and final operations of the unit are
grouped, respectively, in the ExistingGroup and ExpandGroup. And finally, the “Compatibility
Data (Phasing, Prohibiting, Purging, Postponing)” section is where the relationship of the
Commission stage or mode treated as a Purging is configured.

32. 32
Fig. 5. Gantt chart for expansion of a generalized CIP example.
From the Gantt chart in Fig. 5 we can verify the timing for the Commission and Expanded unit-
operation on the Process unit. The black horizon bar means that the binary variable of the shape
is active. The interesting detail is the capacity pool holdup or inventory trend of capacity
(Capacity). We can see a charge of capacity (Charge) in time-period one, and a continuous draw
or dispatch out in time-periods two and three. These values can be seen in Fig. 6.

33. 33
Fig. 6. Amounts of capacity-flow in the GCIP formulation of the motivating example 1.
It should be noted in the constraint instances (supplementary material, lines 183-365) that when
lower or upper bounds of the streams, units or their selection are zero, Eqs (16)-(27) for these
bounds are not constructed. When the bounds are the same, these inequality constraints are
reduced to equality constraints (see lines 234-245 and lines 255-265 in the supplementary
material). Other equality constraints are related to the flow balance in the tanks (lines 269-271)
and to some sequence-dependent constraints (lines 308-323 and lines 340-342). In lines 275-277,

34. 34
330-335 and 343-345 there are the SOS1 (special order set) integrality condition used
the branch-and-bound search of the MILP and provides more intelligent way to solve the
optimization problem by helping to speed up the search which is well known (Williams, 1999).
They are in the list of constraints reported in IMPL’s results, although they are not considered as
constraints by themselves.
4.3. Motivating Example 2
An installation structure similar to the expansion in Fig. 3 is added to the motivating example 2.
The Existing, Commission and Expanded modes or stages case is modified to NonExisting,
Construction and Installed modes for the installation as shown in Fig. 7.

35. 35
Fig. 7. Motivating example 2: small GCIP flowsheet for expansion and installation.
The problem is solved as an MILP using with an optimal objective function of $5.00. If we apply
no expansion or installation then the profit would be $3.0 since the existing capacity is 1.0 for
three time-periods. To perform an expansion, we have the same as the in motivating example 1.
To perform an installation of the same 1.5 quantity-units, the capital cost required is 1.5 * 0.5 +
0.5 = $1.25. Given the timing of the Construction stage, this implies that the Installed stage
occurs in time-periods two and three, which is enforced by the sequence-dependent setup
modeling, i.e., after the Construction stage only the Installed can be setup for the rest of the
horizon. Different from the expansion cost evaluation, there is no existing capacity for an

36. 36
installation (NonExisting mode), so the term “alpha*oldcapacity” is not discounted in the beta
cost (see line 97 in the supplementary material).
In the example, the production from the Construction stage was disregarded by considering the
out-port linked to the product perimeter B with zero yield. With an expansion capital cost of
$0.75 and an installation capital cost of $1.25, a revenue for the sale of material B of 1.0 + 1.5 +
1.5 = $4.0 from the expanded existing unit and 0.0 + 1.5 + 1.5 = $3.0 from the installed non-
existing unit for the three planning periods. This leaves a profit of $4.0 + $3.0 - $0.75 - $1.25 =
$5.0, which is the same value found by the MILP. The Gantt chart for this example is found in
Fig. 8. To permit production during all time-periods, the Commission stage in Process1 during
the first time-period has the same process capacity as in the Existing mode that is in fact a flow
xmfm,t internally in the IMPL modeling of the example. Similarly to Fig. 6 in the motivating
example 1, the amounts of capacity-flow in motivating example 2 considering the GCIP
formulation proposed in the work has the same profiles for charge-size (Charge1 and Charge2)
of the giving capacity-flow with value of 1.5 quantity-units to be filled in the capacity-flow tanks
(Capacity1 and Capacity2) with holdups of 3 quantity-units enough to maintain 1.5 quantity-
units in the second and third time-periods in the Expanded and Installed modes of operation.

37. 37
Fig. 8. Gantt chart for expansion and installation of a generalized CIP example.
5. Examples
Two capital investment planning examples include (i) a retrofit problem for expansion and
extension of 3 process units where investment costs are considered coefficients as the
modifications are pre-defined as fixed values, and (ii) an expansion and installation problem for
an oil-refinery with variable investment costs since the size of the revamped (expanded or
installed) capacity can vary. The examples were modeled using Industrial Algorithm’s IMPL and
solved using IBM’s CPLEX 12.6.
5.1. Retrofit Planning of a Small Process Network

38. 38
Our illustrative retrofit example is taken from Jackson and Grossmann (2002) where there are 3
feeds (A, B, C) and 2 product materials (D, E) with 3 processes (Fig. 9) that can be either
expanded (capacity increase) or extended (conversion increase) or both considering these
modifications as fixed values.
Fig. 9. Retrofit example (Jackson and Grossmann, 2002) for capacity (expansion) and capability
(extension) projects.
In the example, three time-periods are considered. Fig. 10 is equivalent to Fig.9 in our new
flowsheet representation with the Existing, Commission and Expanded (or Extended) stages
shown explicitly where M denotes the flow of money (investment costs). Each process unit (1, 2,
3) has 3 types of modifications (increased capacity, increased capability, both) considering
commission stages (Commission1, Commission2, Commission 3) and their respective final
stages (Expanded1, Expanded2 and Expanded3). Expanded1 represents increase in capacity.
Expanded2 is increase in conversion (capability) and Expanded3 are both. These types of
modifications are only coefficients in the problems in both material balances and costs
evaluation. In the case, for capacity increase, Process 1 can increase from 50 to 75 tons/day,
Process 2 from 85 to 105 tons/day and Process 3 is from 65 to 80 tons/day. For conversion,

39. 39
Process 1 increase from 0.9 to 0.95. Process 2 from 0.85 to 0.90 and Process 3 is from 0.80 to
0.85.
Fig. 10. UOPSS flowsheet for Jackson and Grossmann (2002) example.

40. 40
The associated MILP model using our approach in IMPL includes 929 constraints, 442
continuous variables, and 251 binary variables, and was solved with CPLEX to optimality in less
than 0.25-seconds. The convex-hull MILP formulation proposed in Jackson and Grossmann
(2002) has 394 constraints, 244 continuous variables, and 36 discrete variables, and was solved
in less than 0.3-seconds. Although the larger number of variables and constraints found in our
proposed model when compared with the Jackson and Grossmann (2002) model, the GCIP
formulation using the UOPSS modeling and sequence-dependent setups is solved in smaller CPU
time.
The two Gantt charts shown in Figs. 11 and 12 have the same solution with an NPV = M$
11.841 (M$ in millions of U.S. dollars) over three one-year time-periods where Process 1 has its
capacity expanded, Process 2 continues unchanged and Process 3 has its capability (conversion)
extended. Fig. 11 shows the typical project chart without considering stages of the projects. In
Fig. 12, as proposed in the GCIP modeling, an initial stage of one time-period is regarded for the
3 types of modifications depicted in Fig. 11. To compare both CCIP (Fig. 11) and GCIP (Fig. 12)
NPV, the throughputs and yields of the Commission and Expanded modes in the GCIP modeling
are considered the same.
Fig. 11. Gantt chart of the Jackson and Grossmann (2002) example (CCIP modeling).

41. 41
Fig. 12. Gantt chart of the Jackson and Grossmann (2002) example reformulated as a GCIP.
It should be noted that the results of this retrofit example are different than the ones reported in
Jackson and Grossmann (2002), since these authors reported incorrect numbers in their Figure 3,
although their original GAMS file was correct, and is in fact the one we have used in this paper
to represent in Fig. 11.
5.2. Oil-Refinery Process Design Synthesis
The oil-refinery example in Fig. 13 represents a complete plant with expansions and installations
allowed only for the crude-oil (atmospheric) and vacuum distillation units (CDU and VDU) and
diesel hydrotreating (DHT) over three time periods. The capacity upper bound of the remaining
units are considered a large number to avoid bottlenecks. The Commission/Expanded and the
Construction/Installed structures or shapes for DHT (expansion) and DHT2 (installation) are

42. 42
depicted in Fig. 14. They are constructed similarly to CDU and VDU (expansions) and CDU2
and VD2 (installations). The resulting MILP model involves 2005 constraints, 729 continuous
variables, and 684 binary variables, and was solved with CPLEX in 0.5-seconds, and solved to
optimality yielding a NPV = 4,285 M$. In the example, the total time period is 15 years with 3
periods of 5 years each. A similar study considering expansions and installation of oil-refinery
units can be found in Menezes, Kelly and Grossmann (2015) where nonlinearities to determine
yields and rate of conversions are considered in an iterative decomposition involving MILP and
NLP problems until convergence is achieved. They modeled crude-oil as pseudo-components,
micro-cuts or hypotheticals as found in Menezes, Kelly and Grossmann (2013) and Kelly,
Menezes and Grossmann (2014) to meet an optimal crude-oil slate or diet. A future work
involving the proposed GCIP formulation by using project stages with the MILP + NLP
decomposition and crude-oil as pseudo-components will be proposed in a near future.

43. 43
.
Fig. 13. Oil-refinery example network.

44. 44
Fig. 14. Gantt chart for DHT expansion and DHT2 installation.
Figs. 15-17 show the capacity-flow profiles of the results, where only the installations CDU2,
VDU2, and DHT2 the installations are turned-on or approved. In the example, we consider the
Commission stage of CDU, VDU, and DHT with zero yield, so we expect only installation
project setups. The capacity-flow charge of these projects occurs with their constructions, and at
the same time the capacity-tank is filled to maintain capacity-flow continuously for the Installed
mode as seen in Figs. 15-17. The capacities of CDU, VDU, and DHT in 103 m3/h are 1.375, 0.7,
and 0.4, respectively. These values in the IMPL configuration of the problem are the upper
bounds of the flow in the units xmfm,t. The allowed capacity expansions or unit flow upper
bounds are 20% of the existing capacities, and the allowed installations or unit flow upper
bounds at the Installation modes are 2.0, 1.0, and 0.6 in 103 m3/h for CDU2, VDU2, and DHT2,
respectively.

45. 45
Fig. 15. Amounts of capacity-flow in the GCIP formulation of the installation CDU2.

46. 46
Fig. 16. Amounts of capacity-flow in the GCIP formulation of the installation VDU2.

47. 47
Fig. 17. Amounts of capacity-flow in the GCIP formulation of the installation DHT2.
6. Conclusion
In summary, we have proposed a generalized network-flow MILP model using a sequence-
dependent setups approach to model and solve capital investment planning problems considering
intermediate stages between the existing state (existing unit or new unit) and the final state after
the repair, retrofit or revamp (expansion, extension and installation) projects. We have applied

48. 48
the concepts of sequence-dependent setups to manage the project scheduling by considering
phasing (from one state or stage to another) and the project execution, similar to the repetitive-
maintenance or purging stage as normally found in scheduling sequence-dependent changeover
or repetitive-maintenance problems. A unique and novel way is used to formulate the CIP
problem using capacity/capability and capital as flows in a scheduling environment. Finally, our
generalized CIP formulation can be straightforwardly applied to any CIP problem found in many
process industries, including specific applications such as shale-gas well startup and ore-mining
and extraction planning, which require more scheduling details to be considered.
Author information
Corresponding Author
*E-mail: jdkelly@industrialgorithms.ca
Index Names:
u: units
m: unit-operations
i: unit-operation-port-states (in)
j: unit-operation-port-states (out)
sg: unit-operation sequence-groups
t: time-periods (ntpf: in both the past and future time-horizons. ntf: time-periods in future time-
horizon)
Paramaters Names:
LBfm: lower bound for flow f of the unit-operation m
LBfj,i: lower bound for flow f of the stream between j and i
LByi,m: lower bound for yield of the in-port i of m
LBym,j: lower bound for yield of the out-port j of m
UBfm: upper bound for flow f of the unit-operation m
UBfj,i: upper bound for flow f of the stream between j and i

49. 49
UByi,m: upper bound for yield of the in-port i of m
UBym,j: upper bound for yield of the out-port j of m
Variable Names:
v2r_ymsum,ntpf:unit-operation m setup variable (binary).
v3r_yjisuj,i,ntpf:unit-operation-port-state-unit-operation-port-state ji setup variable (binary).
v2r_xmfm,ntpf:unit-operation m flow variable.
v2r_xmhm,ntpf:unit-operation m holdup variable.
v3r_xjifj,i,ntpf:unit-operation-port-state-unit-operation-port-state ji flow variable.
v2r_zmsum,ntpf:unit-operation m startup variable.
v2r_zmsvm,ntpf:unit-operation m switchover-to-itself variable.
v2r_zmsdm,ntpf:unit-operation m shutdown variable.
v2r_zysgsusg,ntf: unit-operation sequence-group sg memory setup variable.
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