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Journal of Applied Operational Research (2018) Vol. 10, No. 1, 2–24 ISSN 1735-8523 (Print), ISSN 1927-0089 (Online)
www.orlabanalytics.ca
An MILP production scheduling model for a phosphate fertilizer plant using the discrete time representation
Lara Cristina Alves da Fonseca 1,2
, Valéria Viana Murata 1, and Sérgio Mauro da Silva Neiro
1, 1 School of Chemical Engineering, Federal University of Uberlândia, Brazil 2 Votorantim Metais, Companhia Brasileira de Alumínio, Brazil
Received 01 February 2017 Accepted 09 June 2017
Abstract—Fertilizer industries have a strategic importance for the intensification of agriculture and replen-
ishment of soil nutrients that are required to meet the food demand of the growing world population. The
application of production scheduling is one of the ways to increase the operational efficiency of such
plants. The main aim of this research is the development of an optimization model for the production
scheduling problem of a typical phosphate fertilizer plant. The formulation is based on the discrete time
representation and the ability to cope with the inherent features of the fertilizer industry is evaluated,
namely: multipurpose plant comprised of continuous and batch processes in which the batch steps are
characterized by having excessive long processing times; mixed inventory policies; sequence-dependent
changeover; due dates and satisfaction of restrictive operating rules. The production schedule was represented
by an MILP (Mixed integer linear programming) problem considering a scheduling horizon of 30 days.
Three case scenarios were evaluated considering different aspects of the business environment and plant
operations. Solution time showed to be dependent on time granularity but despite the problem dimension,
the proposed discrete based formulation was able to successfully produce programs where detailed operation
was obtained in reasonable time allowing for schedulers to use the proposed model as an effective decision
making tool.
Published online 05 January 2018
Copyright © ORLab Analytics Inc. All rights reserved.
Keywords:
Mathematical modeling
Optimization
Phosphate fertilizers
Production
Scheduling
Introduction
The fertilizer industry has received special attention because of the world population growth and the associated increase in
food demand, expanding biofuel production, and the reduction of arable areas. A production rate adequate to meet world-
wide food demand requires fertilizer usage. Dawson and Hilton (2011) estimate that in 2050, only half the population will
be fed if global fertilizer production does not rise until then. According to Loureiro et al. (2005), the lack of phosphate
fertilizers in soil cultivation directly affects the vegetable growth and development rate, leading to reduced crop yields.
Typical phosphate fertilizer production processes involve continuously operating units and batch stages. Phosphate rock
is the primary raw material for the production of intermediate products, which in turn are used for the production of
numerous end fertilizer types. The problem includes intermediate storage, shared production units, and setup times involved
in product exchange within the same unit, thereby increasing operational complexity.
Recently, some optimization studies have been published, focusing on specific aspects of the fertilizer production process.
Mangwandi et al. (2013) addressed cyclone operation optimization for the production of concentrated phosphate rock,
while Abdul-Wahab et al. (2014) studied the granulation step. Academic papers on production scheduling describing
fertilizer phosphate production operations are not sufficiently detailed. In general, scheduling is based on the scheduler’s
experience or simple heuristics, which overlook important operational process constraints. The main aim of this research
is the development of an optimization model applied to the production scheduling problem of a typical phosphate fertilizer
plant. The presented formulation is a modified version of the model proposed by Kondili et al. (2003). The model is
based on the discrete time representation and the ability to cope with the inherent features of the fertilizer industry
is evaluated, namely: multipurpose plant comprised of continuous and batch processes in which the batch steps
Correspondence: Sérgio Mauro da Silva Neiro, School of Chemical
Engineering, Uberlândia Federal University / UFU, Av. João Náves de
Ávila, 2121, Campus Santa Mônica, Bloco 1K225 – 38408-144
Uberlândia/MG, Brazil
E-mail: [email protected]
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are characterized by having excessive long processing times; mixed inventory policies (unlimited intermediate
storage, UIS, finite intermediate storage, FIS, and non-intermediate storage, NIS); sequence-dependent changeover; due
dates compliance and satisfaction of restrictive operating rules that will be detailed later on in the text. The discrete
time representation was selected for its simplicity and for its ability to produce tight formulations, although they are
known to usually result in rather large problems, which is very dependent on the scheduling horizon length and the required
time grid accuracy.
The text is organized as follows; Section 2 presents the literature review. Section 3 discusses the various phosphate
fertilizer production processes, followed by the description of the plant under study. Section 5 explains the production
scheduling model. Section 6 presents the results and discussion. Finally, the conclusion are drawn in section 7.
Literature review
Following is an outline of the most relevant developments presented in the literature related to the present work, followed
by a discussion on pros and cons of using the discrete time representation.
Evolution of production scheduling problems
Until 1993, production scheduling was applied to low-complexity problems of the manufacturing industry. In the same
year, Kondili et al. (1993) proposed a simple scheduling model for short-term multipurpose batch plants with discrete
time representation. The model was able to handle various scheduling criteria such as flexible equipment, variable batch
size allocation with a fixed processing time estimate, variable utility consumption during batch processing time, and ability
to handle different inventory policies: UIS, FIS, NIS, and zero wait (ZW). Furthermore, the model is able to identify
changeovers, even with several unallocated intermediate periods between tasks, or simply impose sequence-dependent
changeover without identifying transition. The objective function targets revenue maximization deducted from raw material,
inventory, and utility costs. The model was based on a representation the authors named STN - State Task Network. The
originally proposed model involves allocation constraints, which generate poor relaxations, which can be improved, as
shown by Mendez et al. (2006). Besides the modeling aspect, the solution performance of the MILP model proposed by
Kondili et al. (2003) is too dependent on modeling as well as on the computational resources and solution algorithm used.
Mendez et al. (2006) report the combined effect of modeling, high-performance computers, and algorithm solvers containing
more built-in intelligence, while solving an illustrative problem on the work of Kondili et al. (1993), at three time points:
1987, 1992, and 2003. The total reduction of computational time between the early and later years was several orders of
magnitude, from 908 s while exploring 1466 nodes to only 0.45 s when exploring 22 nodes. This shows that models based
on discrete time representation still have practical application potential.
Pantelides (1994) opted for a representation similar to STN, but resource-based instead of state-based. The RTN repre-
sentation (Resource Task Network) extends the proposal of Kondili et al. (2003) incorporating renewable resources to the
representation (i.e., processing units). Thus, this representation explicitly presents which units are used to develop tasks in
a plant. STN and RTN representations were used as a base for the development of various models reported in the 90s
literature and still used to date. The main difference between STN-based models and RTN-based models lies in the way
the resource balance constraints are written, which is more generic in the RTN-based models.
Several other studies innovated and improved time representation, processes, and resolution methods. Ierapetritou and
Floudas (1998) presented a novel STN mathematical formulation for the short term scheduling of batch plants, which was
extended to address continuous and semi continuous plants (Ierapetritou and Floudas, 1998b). In both works, the authors
introduced the non-uniform time grid, also known as an event-based, unit-specific or asynchronous representation. In this
kind of representation, there is a sequence of event point instances located along the time axis of a unit, each representing
the beginning or ending of a task. The location of the event points is different for each unit, allowing different tasks to
start at different times in each unit and producing heterogeneous time grids across different units. Besides the introduction
of the multiple time grid, decision variables regarding task-to-unit allocation were decoupled in two distinct kind of binary
variables with the purpose of reducing model size, a statement that have been disproved by Sundaramoorthy and Karimi
(2005). Later, Ierapetritou et al. (1999) improved their previous formulations allowing demand to be spread along the time
horizon and requiring the need to deal with due dates.
Fonseca et al (2018)
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The most complicated aspect of formulations based on a non-uniform time grid is the fact that they must be able to
efficiently model interactions of producing and consuming tasks involving a common intermediate material state.
Inconsistencies in material balances and violation of time horizon constraints may arise, as pointed out by Castro et al.
(2001), Maravelias and Grossmann (2003) and Sundaramoorthy and Karimi (2005). In order to address the inconsistency
in material balances, storage must be represented as separate tasks in the Ierapetritou and Floudas’s model. Janak et al.
(2004) presented an enhanced formulation with respect to that presented by Ierapetritou and Floudas (1998) in which tasks
are allowed to take place over multiple event points and thus overcoming the inconsistencies in material balance. The proposed
formulation is also able to address mixed storage policies as well as resource constraints. Previously to that work,
Maravelias and Grossmann (2003) proposed an MILP formulation for the short-term scheduling of STN multipurpose
batch plants featuring the same capabilities but using a uniform time grid. The model of Janak et al (2004) and Maravelias
and Grossmann (2003) share the idea of establishing material balances and allocation between event points of tasks that
are performed in units over multiple event points. The introduction of linking variables and constraints cause a significant
impact on the model dimension. A more compact formulation was proposed by Castro et al. (2004) for batch and continuous
processes, in which variables explicitly bore the information on the event a task was let to start and a later event it was finished.
In this case the RTN representation was used in combination with the uniform time grid, which favored minimal use of
Big-M constraints. The proposed approach was an extension of the formulation proposed by Castro et al (2001). In the
improved approach, timing constraints for tasks that shared the same unit were combined in a single constraint under the
assumption that only a task could take place in a unit at a time, instead of treating them individually. The combined constraint
generally produced better relaxation. An important additional parameter of this approach was the maximum number of
slots over which a task was allowed to take place, which required more steps in determining the optimal solution. Following
the same idea as the previous works, Sundaramoorthy and Karimi (2005) proposed a slot-based formulation for the scheduling
of multipurpose batch plants using generalized recipe diagram as an alternative for process representation and allowing
tasks to continue processing over multiple time slots. The authors urge that their novel idea of establishing balances in
terms of time, mass and resources led to a model that used no Big-M constraints.
Still as an effort of circumventing the inconsistencies of the Ierapetritou and Floudas approach, Giannelos and
Giogiadis (2002) proposed an STN formulation using the non-uniform time grid for short-term scheduling of multipurpose
batch plants in which buffer time was added to tasks durations in order add more flexibility to the model. However, the
authors also introduced duration and sequencing constraints that ended up denoting a global event effect to the resulting
model. Because the end time of producing tasks and the start time of consuming tasks were forced to coincide for material
balance and storage constraints purposes, suboptimal solutions were obtained.
Shaik et al. (2006) conducted a comprehensive comparative study of formulations found in the literature for the scheduling
of multipurpose batch plants including the formulations proposed by Castro et al (2001), Castro et al (2004), Giannelos
and Giordiadis (2002), Maravelias and Grossmann (2003), Sundaromoorthy and Karimi (2005) and a modified version of
Ierapetritou and Floudas (1998). A collection of benchmark problems ranging from small to medium size were used to test
statistical and computational performance. Both optimization directions were considered maximization of profit and
minimization of makespan, the latter being considered to be a more difficult kind of optimization problem. As a general
conclusion, the modified version of Ierapetritou and Floudas produced models with smaller dimensions due to the fact that
it required less time slots, besides producing better relaxation solutions and the least computational times. In some cases,
the formulation of Giannelos and Giordiadis (2002) was not able to determine the optimal solution whereas the approach
of Castro et al (2001), Castro et al (2004), Maravelias and Grossmann (2003) and Sundaromoorthy and Karimi (2005)
resulted in larger models that consumed longer computational times. For the latter approaches, in many instances, the
terminating criterion was attaining a maximum solution time, in which case the relative gap was not closed. Generally
speaking, non-uniform time grid models require less event points compared to the corresponding global-event or slot-based
models, thus yielding better computational results. On the other hand, non-uniform time grid models usually make use of
Big-M constraints in building timing constraints.
Shaik and Floudas (2007) proposed an improved approach for the short-term scheduling of continuous processes
considering rigorous treatment of storage requirements. The same authors (Shaik and Floudas, 2008) also proposed a RTN
version of the improved STN version of Ierapetritou and Floudas presented in Shaik et al. (2006).
Pros and Cons of the discrete time representation
According to Floudas and Lin (2004), the advantage of using discrete time representation is that time grids are used as a
reference for all operations competing for shared resources. A common time grid with predefined points, where operations
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can be initiated or completed, allows for straightforward and simple model construction. Rapidly increasing problem sizes
is due to time grid refinement to address different operation durations, which is its main disadvantage. Although various
forms of continuous time grids have been proposed in the literature to make models more flexible and smaller, as can be
noted by the literature review discussed in the last section, the use of models based on discrete time representation has still
drawn attention of the scientific community. Velez and Maravelias (2013, 2015) have recently proposed improvements to
the discrete time representation when time refinement is required. Their main idea was to create a discrete non-uniform
time grid that was not only unit-specific but also task-specific and material-specific as well.
Despite the large number of published studies and important contributions identified over the past decades, no model is
generic enough to cover all aspects of all scheduling problems or is one model superior to all others for all problems.
Therefore, distinct model forms must test problems of different natures.
Phosphate fertilizers
Phosphorus, nitrogen, and potassium are three macronutrients essential to any plant survival. Phosphorus is a key element
in the process of converting solar energy into nutrients, oils, and fibers. It is required in photosynthesis, sugar metabolism,
nutrient storage and transfer, cell division, growth, and cell information transfer. Phosphate fertilizers account for over
60% of fertilizer production, with its demand growing 2.4% globally and 4.0% in Latin America. China, Russia, India,
and the United States represent more than 50% of world consumption, thus pricing such products worldwide. The estimated
global consumption is approximately 100,000 ktons/year (Research and Marketing, 2014).
The commercial production of phosphate fertilizers worldwide is based on the exploitation of natural deposits of mineral
phosphatic material, known as phosphate rocks. Given their volcanic origin (not sediments), these rocks are mostly insoluble
in water, making phosphorous absorption impossible by vegetables. To provide phosphor, mine-extracted rock must hence
undergo chemical or thermal processes. The P2O5 content in virgin rock (a measure of phosphorus amount in rock) usually
varies between 2% and 22% (Kulaif, 2009). The P2O5 content in phosphatic rocks can be increased. Phosphate concentrate
can be obtained when subjected to the following steps: screening, water addition, hydrocycloning, calcination, flotation,
and magnetic separation (IPNI, 2015). The most marketed raw materials for phosphate fertilizer production are rock
phosphate concentrate with 33%–38% P2O5, sulfuric acid, phosphoric acid, lime, and ammonia. Figure 1 schematically
illustrates the transformation process of raw materials into finished products. The production of such fertilizers is widely
known, and its production processes are identical worldwide (Cekinski et al., 1990).
Figure 1. Phosphate fertilizers: main products (adapted from INPI, 2015).
Reaction (1) between phosphate rock with concentrated sulfuric acid and water in the stoichiometric reaction produces
agriculturally used phosphogypsum (CaSO4.nH2O), hydrofluoric acid, and phosphoric acid. The latter is also used in the
production of phosphate fertilizers.
Ca10F2(PO4)6 + 10H2SO4 + 10H2O → 10CaSO4.nH2O + 6H3PO4 + HF (1)
Various products can be generated from the same reactants and equipment. What makes one product different from
another is the reactant’s proportions. The main products sold are as follows:
Phosphate Rock
SSP Phosphoric Acid
TSP MAP SSPA
+H2SO4 +H2SO4
+ NH3 + NH3 + phosphate rock
CaSO4
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Figure 2. Fertilizer phosphate production flowchart.
Simple super phosphate (SSP) - CaH(PO4).2H2O: It is a fertilizer with a low phosphorus concentration. It is the most
important fertilizer used for blending with other secondary nutrients. Its production results from a slow reaction (2) between
concentrated phosphate rock, sulfuric acid, and water and takes days to finish completely. The reaction is initiated with
acidulation and transferred to warehouses that are sub dived in stalls where the resultant solid product rests until the reaction
is completed: a stage conventionally known as curing.
Ca10(PO4)6F2 + 7H2SO4 + 6.5H2O → 3CaH(PO4).2H2O + 7CaSO4.½H2O + 2HF (2)
Simple superphosphate ammoniated (SSPA): It is a fertilizer made by mixing SSP with ammonia. It should contain 1%
of bulk nitrogen.
Super triple phosphate (TSP) - CaH4(PO4).H2O: It is a phosphatic fertilizer with a high phosphorus content. It is produced
by reacting phosphate rock, phosphoric acid, and water (3). Similar to the reaction involved in SSP, TSP production is
quite slow, thereby requiring curing.
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Ca10(PO4)6F2 + 14H3PO4 + 10H2O → 10Ca(H2PO4)2.H2O + 2HF (3)
Monoammonium phosphate (MAP) - NH4H2PO4: It results from a reaction between phosphoric acid and ammonia. It is
the most frequently sold product.
NH3 + H3PO4 → NH4H2PO4 (4)
The block diagram in Figure 2 shows a generic phosphate fertilizer production process. Because of large fertilizer
consumption, production plants require auxiliary plants to produce sulfuric and phosphoric acid used as inputs to fertilizer
production. Sulfuric acid is produced from sulfur and water. Involved reactions are highly exothermic and act as steam
and electricity generators. Sulfuric acid is used in the manufacture of phosphoric acid by reaction (1). After the reaction,
acid is concentrated with an evaporator (the final concentration depends on its application). Concentrations used for reaction
(1) and reaction (4) are different. With the evaporator being shared during phosphoric acid production for both applications,
its use must be programmed to avoid frequent exchanges and thereby the production of off-spec materials.
SSP and TSP acidulation steps involve reactions (2) and (3), respectively. After the reaction, produced materials are
stored in warehouses for a few days until reactions are complete: a step identified as curing.
During this process, phosphatic concentrate solubilization occurs upon reaction completion. Hakama et al. (2012) have
shown that the longer the curing, the better is the P2O5 solubilization. The recommended minimum time between acidulation
reaction and soil application is 5 days.
Cured SSP and TSP result in intermediate solids that need to be granulated to reach its end product form. In this SSP
and TSP granulation process, calcium is added as input to generate products that also require curing before being
commercialized. SSP can also be granulated with the addition of ammonia to increase the end compound’s nitrogen content.
MAP is produced by reaction (4), with phosphoric acid and ammonia as inputs.
Problem statement
For the production scheduling problem for a generic phosphate fertilizer plant, we selected four of the main
products mentioned in the previous section: SSPA, SSP, TSP, and MAP. The flowchart in Figure 3 illustrates the
case study’s production flowchart.
Figure 3. Case study: block flow diagram.
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Figure 4. STN representation of the case study phosphate fertilizer plant.
The phosphate fertilizer process considered requires two additional plants: one dedicated to sulfuric acid production and
another dedicated to phosphoric acid production (both are raw materials used in fertilizer production). Fertilizer production is
initiated in the acidification units where phosphate rock is put in contact with acids. A single unit is availab le for the
intermediate product production (uncured SSP and TSP). These two products cannot be produced simultaneously given
the different unit configuration required by each. In addition, the recipe is different for each product. The reaction results
in SSP when using sulfuric acid, whereas a reaction between phosphoric acid and phosphate rock produces TSP. The
production schedule should thus indicate how to use the unit in an effective manner, avoiding frequent product exchanges
and satisfying demands.
After acidification, intermediate products should rest until cure is complete. Curing solid materials are stacked on dedicated
warehouses and divided into stalls organized by various maturation stages. Each warehouse has five stalls. Warehouse 1 is
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dedicated to SSP cure with a 5-day curing time, whereas warehouse 2 is assigned to TSP cure, with a 6-day curing time.
The different reaction times are because SSP uses sulfuric acid (which reacts quickly due to its acid strength), whereas
TSP requires a higher percentage of soluble phosphorus in the final product. Each stall has a maximum and minimum
amount to be stacked prior to materials’ curing. Curing only starts if stacks have not been loaded with new acidification
batches. After the completion of the curing period, materials remain stored in stalls until transferred to granulation units.
Stalls can only receive new material to be cured after previous cure is completely finished to prevent mixing cured material
stacks with uncured material batches. Therefore, stalls can at any point in time be empty, receive material to form stacks,
or have stacks of curing or cured materials to be transferred to granulation until their total consumption.
Granulation units transform intermediates into final products. The plant has three granulation units with different capacities
are able to produce four types of final products. However, shifting from one product to another requires equipment
reconfiguration and hence a setup time.
Reaction (4), between phosphoric acid and ammonia, produces MAP, the only product without curing and thus it is the
only product that is able to be produced continuously. Nitrogen is added to SSP for SSPA production to achieve the required
nitrogen content. Cured lime or TSP is added to some end products to impart the contents prescribed by each product’s
legislation. Such amounts can vary as required if desired acidification levels are not met. After granulation in production
units, both SSP and TSP remain in cure for an additional day until released for shipment. This process prohibits equipment
production exchanges during weekends because of lack of supervisory/monitoring activity.
Figure 4 represents the plant described above by the STN representation. Gray circles indicate raw materials, white circles
indicate intermediate states, and black circles depict end products. Infinite storage capacity is assumed for raw materials
and final products (UIS). Dedicated tanks of limited capacity store sulfuric acid and the various phosphoric acid types
(FIS). Once curing in stalls, SSP and TSP should remain there until resources are fully consumed. Other storage resources
for intermediate products are nonexistent (NIS). Boxes represent operations, while colored rectangles express operations
sharing equipment (corresponding to phosphoric concentration units, acidulation units, and granulation units 1, 2, and 3).
Phosphoric acid concentration unit can produce phosphoric acid of a suitable concentration for both TSP and MAP
production. The acidification unit is also shared and is considered a major process bottleneck, given that the production of
three out of four products need the intermediate products generated in this unit. The changeover time to swap from SSP to
TSP production is 8 h and vice versa.
Granulation units are capable of processing any product type. There are three parallel units. In each, only one product
type is produced per time interval. A cleaning time is associated with the exchange between each two different products
within a single unit, in view of a possible modification of components content dictated by regulation, even if there is just a
small quantity of product remnant.
Gray rectangles map different stalls of a single warehouse and may be considered parallel units. Cure takes place in
stalls; this activity demands greater focus because of the long processing times. Tables A1-A6 in the Appendix present
unit process capacities, batch processes data, initial inventory, changeover time, gross profit and consumption factors,
respectively.
According to classification presented by Mendez et al. (2006), the fertilizer production process just discussed can be
classified as a process network with inventory policy encompassing UIS, NIS and FIS states. Demand must meet due
dates, and there is the incidence of sequence-dependent changeovers. Production units may have variable loads; however,
their process times remain load independent.
Mathematical modeling
From the STN representation of a phosphate fertilizer plant, an optimization model was constructed inspired by the proposition
of Kondili et al. (1993). The model uses the following nomenclature:
Index
i Tasks
j Production units
s States (raw materials, intermediate products, or end products)
t Time frames
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Sets
Task i consuming state s
Task i producing state s
States s consumed by task i
States s produced by task i
Ij Tasks i that can be performed in unit j
Jb j units running batch jobs
Jc j units performing continuous tasks
JCO
j units with changeover
Ji j units able to perform task i
Scc
States s produced by continuous processes and consumed by continuous processes
Sbc
States s produced by batch processes and consumed by continuous processes
Scb
States s produced by continuous processes and consumed by batch processes
Sp States s corresponding to end products
Srm
States s corresponding to raw materials
Parameters
Batch job i, Maximum load
Batch job i, Minimum load
Maximum load of continuous task i in unit j
Minimum load of continuous task i in unit j
Unit j, Maximum storage capacity
Unit j, Minimum capacity storage
Set up time in going from task i′ to task i
State s consumption factor for task i
State s production factor for task i
Dems State s, Demand
K Changeover Penalty
Ms Product s, Contribution margin
Batch tasks I, Processing time
Continuous variables
Task load batch i in unit j in time t
Continuous task i load in unit j in time t
Ajst State s quantity kept in stall j in time t (after curing)
Indicates needs of changeover from task i to task i' in time t
CM Objective function value
Dst Product s quantity used to meet demand in period of time t
Fjst State s quantity accumulated in stall j in time t (before curing)
Lst State s quantity kept in inventory in time t
Pst Purchase quantity of state s in time t
Binary variables
Wi,j,t Indicates whether unit j begins processing task i at time t
Xjt Indicates the presence of uncured material accumulated in stall j in time t
Yjt Indicates the presence of cured material stored in stall j in time t
Zi,j,t Indicates whether no tasks are assigned to unit j in time t with task i being the last task performed
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For continuous operations, the processing time is equivalent to one period of time. Should an operation extend
over several periods of time, allocation should be established sequentially in all periods during which the activity
occurs. In contrast, batch operations (cures) are started in a period of time and extend over various time periods,
in which case, allocation should only occur in the time period in which a task is initiated.
Task allocation to process units: For process units capable of developing numerous tasks, only one task can be
allocated at a time. Constraints (5) are used for continuous operations, while constraints (6) are used for batch
operations.
(5)
(6)
Setup time between two different products: These constraints ensure that the unit exchange setup time between
task i' and i is respected, where i, i’ .
(7)
Changeover ID: In addition to imposing setup times, changeovers should be identified and penalized to prevent
frequent recurrence. Changeover is identified through the set of constraints (8), which are introduced by the present
work. The use of the original constraints proposed by Kondili et al. (2003) would be prohibitive in industrial
size problems given that their constraints would unfold in a huge number of individual constraints as a result of
having all (t, t’) combinations with t’ > t. In order to circumvent that problem the idea was to use a constraint as
simple as (8a) without variable on the right hand side of the inequality. However, if that constraint was
used, idle time periods would be created between consecutive time periods resulting in the identification of
changeover impossible. Moreover, that constraint would not be suitable for the cases where task dependent
changeover was involved because of the associated variable number of time periods. Therefore, the idea was to
include a variable that would carry the information on the last task executed on the unit when idle time periods
were allocated between different tasks. In order to always be able to have an activated variable at a given time
period, constraints (5) are replaced by (8b), whereas (8c) are required for activating the right auxiliary variable
when an idle time is allocated right after a task or for transferring the information on the last task performed
when multiple idle time periods are assigned. In cases where it is sufficient to ensure an adequate number of
time periods left for cleaning or setting up without penalization, constraints (8) can be suppressed keeping only
constraints (7).
(8a)
(8b)
(8c)
(9)
(10)
Constraints (9) and (10) impose minimum and maximum limits on the quantities of materials processed by
continuous and batch operations, respectively. Constraints (11) state that a material to be cured can only be
placed in a stall if this is allocated for that purpose ( , whereas constraints (12) state that cured materials
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may only remain inventoried in a stall if the latter has also been allocated for that purpose ( ). Note that
material storage accumulation tasks are only associated with resources that develop batch tasks ( ).
(11)
(12)
When curing tasks are allocated to a stall, they should be performed during time periods, corresponding to
cure. Therefore, in this time interval, the stall cannot be allocated to uncured materials or to store cured material,
as guaranteed by constraints (13).
(13)
Constraints (13) are unable to ensure that only one operation occurs per stall between loading and cured material
storage at a given time. Therefore, constraints (14) are added for this purpose.
(14)
Once the stall starts receiving materials to form a stack, allocation for feeding should be continuous until
stacks are complete ( . After the pile has received the final amount of material, the stack starts curing
( ): constraints (15). The amount of accumulated stack material must comply with the minimum and
maximum limits imposed by constraints (10).
(15)
Constraints (13-15) are introduced in this work to be able to efficiently manage loading, curing and storage in
each stall of the warehouses.
The amount of stack material to be cured in the stall can only increase during material accumulation. In constraints
(16), is nondecreased during stack formation unless the stack begins to cure, .
(16)
Similarly, the amount of cured material within a stall can only decrease, preventing new material loads in the
stall until cure is completed and total cured material is consumed. In constraints (17), decreases after curing
task completion.
(17)
Constraints (18) track the total consumption of each raw material type s for individual time frames.
(18)
Constraints (19) denote the mass balance of intermediate products produced by continuous processes and
consumed by continuous processes. Note that contemplates both the intermediate FIS allowed to be
stored ( ) as well as the NIS product ( ). Variable denotes the amount of material intended to meet
the end product’s demand.
(19)
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Constraints (20) establishes the mass balance of intermediate products produced by continuous processes and consumed
by batch processes. The STN representation in Figure 4 shows that (20) are applied to uncured SSP and TSP intermediate
products (which are transferred to any one of the five stalls of its respective warehouses). The equations calculate the
mass balance considering that the total amount of materials accumulated in all stalls ( ) in time t-1 accrued of
total uncured SSP/TSP amounts produced in time t (second term on the constraint’s left hand side), should be equal to the
total amount accumulated in all stalls in time t ( ), accrued by the quantities of materials curing in their stalls
after stack formation (second term on the constraint’s right hand side). Mathematical analyses of equations (20) conduct
to the conclusion that the material stored in stall 1 during a given period could be accounted for in any other stall in the
next period, maintaining material balance consistency but inducing the physical displacement of tons of material between
stalls, which is both undesirable and physically impossible. Material stacks on a given stall will cure in their respective
stall. Model constraints (16) ensure that the material transferred to a stall remains in a single stall until its cure starts.
(20)
Constraints (21) give the mass balance of batch-produced intermediate products consumed by continuous processes.
The STN representation in Figure 4 shows that (21) are applied to intermediate-cured SSP and TSP products (kept cured
and stored in stalls until entirely evacuated to granulation units). These constraints dictate that the mass balance, considering
the total amount of cured materials in all stalls ( ) at time t-1, accrued of total curing SSP or TSP amounts
during time t (second term in the constraint’s left hand side), should equal the total amounts in all stalls in time t ( ,
accrued by material amounts cured in time t (second term on the constraint’s right hand side). Similar to constraints (20),
mathematical analysis of constraints (21) leads to the conclusion that any amount of material stored in a stall in a given
time period could be accounted for in any other stall in the next period, maintaining material balance consistency but
inducing the physical displacement of tons of material between stalls. Constraints (17), in this case, are responsible for
ensuring that the cured material of a stall remains in it until completely evacuated by granulation operations.
(21)
Constraints (22) ensure demand satisfaction for all products at the end of the scheduling horizon. Should there
be need for satisfaction of due dates, (22) are alternatively replaced by (23). The fertilizer industry presents a
seasonal demand with high peaks during the planting seasons. During those periods, demands are usually way
higher than the installed production capacities. Therefore, excess production in off-peak periods are stored to
satisfy the exceeding demand in the planting seasons. That is the reason why constraints (22) and (23) are kept
as inequalities.
(22)
(23)
Objective function: The model’s goal is to maximize contribution margins (first term on the right hand side of
(24)), eventually penalized by the number of changeovers occurring during the scheduduling horizon (second
term on the right hand side of (24)).
(24)
The main purpose of this work is to test the presented formulation, which is a modified version of the model
proposed by Kondili et al. (2003) – (see constraints (8), (13-15) and (24)), against the problem described in the
problem statement. More specifically, the discrete time representation capacities of coping with the inherent
Fonseca et al (2018)
14
features of the fertilizer industry, namely: multipurpose plant comprised of continuous and batch processes in
which the batch steps are characterized by having long processing times; mixed inventory policies (UIS, FIS
and NIS), sequence-dependent changeover, due dates compliance and satisfaction of operating rules that will be
detailed in the next section.
Results and Discussion
The mathematical model resulted in an MILP problem which was implemented in the GAMS system distribution 24.4 using
the off-the-shelf CPLEX distribution 12.6. No tailored solution algorithm was required to solve the problem under study.
The objective function was to maximize the profit at the end of the production schedule horizon. Three different scenarios
were considered to analyze the model’s flexibility, thereby demonstrating the ease of restriction addition (one of the
advantages of discrete modeling representation). The first case namely general model was taken to be a base case with no
additional constraints to those presented in the previous section. The second case added a demand for service restriction
over time (due date). The main purpose of this scenario is to be able to satisfy demand spread over the scheduling horizon.
Another variant of the second case not addressed in this work could be the split of demand into fixed and discretionary.
The third case addressed a product change impediment: granulation and acidulation units barring during weekends due to
lack of supervisory personnel in such periods. By prohibiting changes on weekends, accidents causing property and individual
damages are prevented. The formulation for the two first cases was composed of constraints (5-7), (9-23) and the objective
function (24) without changeover penalization. Changeover penalization was not included in those cases assuming that
once a setup time between different tasks is assigned a lost of production would result. Given that the objective function is
directed to maximize the contribution margin, setup times would naturally be minimized. The third case, on the other hand,
required the addition of the set of constraints (8) with the additional imposition for the time periods corresponding to
weekend days. Moreover, constraints (8a) were imposed only over the periods corresponding to weekend days, whereas
(8b) and (8c) were applied over the entire time horizon yielding a smaller model and better computational performance.
The second and third cases are considered to be more complex in comparison to the first one because the inclusion of
additional constraints apparently turns the problem more restrictive. Table 2 presents model statistics for the three cases.
In all cases, a relative gap of 0 % was used as a stopping criterion guaranteeing to find the global optimal solution.
Table 2. Case studies statistics results*.
General model Due dates Weekends
Relaxed MILP ($) 25,896,000.00 25,896,000.00 25,896,000.00
MILP ($) 25,896,000.00 25,896,000.00 25,896,000.00
Equations 15,636 15,640 18,020
Continuous variables 5,852 5,852 9,452
Binary variables 4,320 4,320 5,760
Number of iterations 135,690 119,557 99,217
Nodes 984 533 502
CPU (s) 52.369 31.964 44.117
Relative gap (%) 0.0 0.0 0.0 *Processor: Intel® Core ™ i7-2860QM CPU @ 2.50GHz (8 GB RAM).
The additional complexity of the second and third cases had no major impact on the computational performance of these
problems leading to the conclusion that the additional constraints actually reduced the feasible region and helped speeding
the process for searching the optimal solution. Comparing case 2 to the general model in terms of problem size, there is
just a small increase in the number of constraints due to the imposition of due dates. When case 3 is compared to the general
model though the impact is more pronounced due to the set of constraints (8) and the introduction of variables and
. However, the computational time is not negatively affected in both cases. It can also observed that the same objective
function value was obtained for all cases. The reason for that is that the end product production profile considering the
whole scheduling horizon was exactly the same in all cases, as can be seen in Table 3. It is clear that MAP production was
Journal of Applied Operational Research Vol. 10, No. 1
15
prioritized. Besides the inexistence of a maximum production constraint for this product, production was quite favorable
given that its process only involved continuous operations with high production rates and high contribution margin. In addition,
resource allocation to maximize TSP production originated higher revenue due to this product’s higher contribution margin
in comparison to other products relying on batches.
Table 3. Demand and production profiles over the whole scheduling horizon.
Product Demand (tons) Production (tons)
General Model
Production (tons)
Due Date
Production (tons)
Weekend
SSP 8,000 8,000 8,000 8,000
SSPA 10,000 10,200 10,200 10,200
TSP 15,000 19,000 19,000 19,000
MAP 20,000 65,000 65,000 65,000
Figure 5. Gantt chart: general model.
The Gantt chart in Figure 5 shows the allocation of tasks to units along the scheduling horizon for the general model. It
depicts stall loading, curing, and storage tasks, strictly following the physical chain of events. I should be noted that both
warehouses presented high utilization with no intermediate product at the end of the scheduling horizon. Synchronization
constraints (such as loading tasks, material cure, and storage in stalls, in a predefined order, and following the criteria of
minimum volume and maximum reaction duration) are better understood through stall details (stall allocation in relation
to material volume and task execution) shown in Figure 6. To be noted for tasks synchronization, curing time of both
15-interval SSP and 18-interval TSP only started to be accounted once material loads respected stipulated quantity minimum
and maximum values. The maximum storage capacity was similarly respected for all production units. Initial materials
cured in stock in stall 1 of warehouses 1 and 2 were also correctly allocated. Consequently, stalls stocking cured material
at the beginning of the production schedule became unavailable until emptied. Uncured TSP production was delayed until
time period 6, when the acidulation unit changes over from SSP to TSP production. The produced material is stacked in
stall 1 of warehouse 2. After curing is complete, the whole cured material is transferred to granulation units 2 and 3 and
another batch is started in the following periods in stall 1. That is why a long continuous green bar is seen in the Gantt
chart. Likewise, the long green bars shown for stalls mean that consecutive batches are processed. No granulation unit was
dedicated, which resulted in the processing of the mix of the end products and involving quite a few changeovers in each
granulation unit. The acidulation unit was underutilized from time period 76 on. The reason for such underutilization lies
in the lack of incentive to extend unit operation beyond that period of time, given that long cures effectively reduce the
time to use intermediate products produced beyond period 76.
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90
H2SO4 ProductionH2SO4H2SO4H2SO4H2SO4H2SO4H2SO4H2SO4H2SO4H2SO4H2SO4H2SO4H2SO4H2SO4H2SO4H2SO4H2SO4H2SO4H2SO4H2SO4H2SO4H2SO4H2SO4H2SO4H2SO4H2SO4H2SO4H2SO4H2SO4H2SO4H2SO4H2SO4H2SO4H2SO4H2SO4H2SO4H2SO4H2SO4H2SO4H2SO4H2SO4H2SO4H2SO4H2SO4H2SO4H2SO4H2SO4H2SO4H2SO4H2SO4H2SO4H2SO4H2SO4H2SO4H2SO4H2SO4H2SO4H2SO4H2SO4H2SO4H2SO4H2SO4H2SO4H2SO4H2SO4H2SO4H2SO4H2SO4H2SO4H2SO4H2SO4H2SO4H2SO4H2SO4H2SO4H2SO4H2SO4H2SO4H2SO4H2SO4H2SO4H2SO4H2SO4H2SO4H2SO4H2SO4H2SO4H2SO4H2SO4H2SO4H2SO4
H3PO4 ProductionH3PO4DilH3PO4DilH3PO4DilH3PO4DilH3PO4DilH3PO4DilH3PO4DilH3PO4DilH3PO4DilH3PO4DilH3PO4DilH3PO4DilH3PO4DilH3PO4DilH3PO4DilH3PO4DilH3PO4DilH3PO4DilH3PO4DilH3PO4DilH3PO4DilH3PO4DilH3PO4DilH3PO4DilH3PO4DilH3PO4DilH3PO4DilH3PO4DilH3PO4DilH3PO4DilH3PO4DilH3PO4DilH3PO4DilH3PO4DilH3PO4DilH3PO4DilH3PO4DilH3PO4DilH3PO4DilH3PO4DilH3PO4DilH3PO4DilH3PO4DilH3PO4DilH3PO4DilH3PO4DilH3PO4DilH3PO4DilH3PO4DilH3PO4DilH3PO4DilH3PO4DilH3PO4DilH3PO4DilH3PO4DilH3PO4DilH3PO4DilH3PO4DilH3PO4DilH3PO4DilH3PO4DilH3PO4DilH3PO4DilH3PO4DilH3PO4DilH3PO4DilH3PO4DilH3PO4DilH3PO4DilH3PO4DilH3PO4DilH3PO4DilH3PO4DilH3PO4DilH3PO4DilH3PO4DilH3PO4DilH3PO4DilH3PO4DilH3PO4DilH3PO4DilH3PO4DilH3PO4DilH3PO4DilH3PO4DilH3PO4DilH3PO4DilH3PO4DilH3PO4DilH3PO4Dil
H3PO4 ConcentrationH3PO4-MAPH3PO4-MAPH3PO4-MAPH3PO4-MAPH3PO4-TSPH3PO4-TSPH3PO4-MAPH3PO4-TSPH3PO4-TSPH3PO4-MAPH3PO4-MAPH3PO4-MAPH3PO4-MAPH3PO4-TSPH3PO4-MAPH3PO4-TSPH3PO4-MAPH3PO4-MAPH3PO4-TSPH3PO4-TSPH3PO4-TSPH3PO4-TSPH3PO4-MAPH3PO4-MAPH3PO4-MAPH3PO4-MAPH3PO4-MAPH3PO4-MAPH3PO4-MAPH3PO4-MAPH3PO4-MAPH3PO4-MAPH3PO4-MAPH3PO4-MAPH3PO4-MAPH3PO4-MAPH3PO4-MAPH3PO4-MAPH3PO4-MAPH3PO4-MAPH3PO4-MAPH3PO4-MAPH3PO4-TSPH3PO4-MAPH3PO4-TSPH3PO4-MAPH3PO4-MAPH3PO4-MAPH3PO4-MAPH3PO4-MAPH3PO4-MAPH3PO4-MAPH3PO4-TSPH3PO4-TSPH3PO4-MAPH3PO4-MAPH3PO4-TSPH3PO4-MAPH3PO4-MAPH3PO4-MAPH3PO4-MAPH3PO4-MAPH3PO4-MAPH3PO4-TSPH3PO4-MAPH3PO4-MAPH3PO4-MAPH3PO4-MAPH3PO4-MAPH3PO4-MAPH3PO4-MAPH3PO4-MAPH3PO4-MAPH3PO4-MAPH3PO4-MAPH3PO4-MAPH3PO4-MAPH3PO4-MAPH3PO4-MAPH3PO4-MAPH3PO4-MAPH3PO4-MAPH3PO4-MAPH3PO4-MAPH3PO4-MAPH3PO4-MAPH3PO4-MAPH3PO4-MAPH3PO4-MAPH3PO4-MAP
AcidulationSSPSSPSSPSSP0TSPTSPTSPTSPTSP0 0 0 0 0 0SSP0 0SSPSSPSSP0TSP0 0TSPTSPTSPTSP0 0 0 0 0 0SSPSSPSSPSSPSSP0 0TSPTSPTSPTSP0TSP0 0 0 0SSPSSPSSPSSP0 0SSP0 0 0TSPTSPTSPTSPTSP0 0SSPSSPSSPSSPSSP0 0 0 0 0 0 0 0 0 0 0 0 0 0 0
Stall 1 (SSP) 0 0 w w w w w w w w w w w w w w w y 0 x x w w w w w w w w w w w w w w w y w w w w w w w w w w w w w w w 0 x x w w w w w w w w w w w w w w w x w w w w w w w w w w w w w w w 0 0 0 0
Stall 2 (SSP) 0 0 x w w w w w w w w w w w w w w w y x w w w w w w w w w w w w w w w 0 0 0 w w w w w w w w w w w w w w w y 0 0 w w w w w w w w w w w w w w w x w w w w w w w w w w w w w w w 0 0 0
Stall 3 (SSP) w w w w w w w w w w w w w w w y 0 0 0 0 x x w w w w w w w w w w w w w w w 0 0 0 x x x x w w w w w w w w w w w w w w w w w w w w w w w w w w w w w w w w w w w w w w w w w w w w w 0
Stall 4 (SSP) 0 x x x w w w w w w w w w w w w w w w 0 w w w w w w w w w w w w w w w 0 w w w w w w w w w w w w w w w y y y y x x w w w w w w w w w w w w w w w 0 w w w w w w w w w w w w w w w 0 0
Stall 5 (SSP) 0 w w w w w w w w w w w w w w w w w w w w w w w w w w w w w w 0 0 0 0 0 0 0 x w w w w w w w w w w w w w w w x w w w w w w w w w w w w w w w y y w w w w w w w w w w w w w w w y 0 0
Stall 1 (TSP) 0 0 0 0 0 w w w w w w w w w w w w w w w w w w w w w w w w w w w w w w w w w w w w 0 0 w w w w w w w w w w w w w w w w w w y 0 x w w w w w w w w w w w w w w w w w w y 0 0 0 0 0 0 0
Stall 2 (TSP) 0 0 0 0 0 0 x x w w w w w w w w w w w w w w w w w w w w w w w w w w w w w w w w w w w w w w w w w w w w w w w w w w w w w w y y 0 0 w w w w w w w w w w w w w w w w w w 0 0 0 0 0 0
Stall 3 (TSP) 0 0 0 0 0 0 0 w w w w w w w w w w w w w w w w w w y 0 0 0 w w w w w w w w w w w w w w w w w w 0 w w w w w w w w w w w w w w w w w w y w w w w w w w w w w w w w w w w w w y 0 0 0 0
Stall 4 (TSP) 0 0 0 0 0 0 0 0 w w w w w w w w w w w w w w w w w w y w w w w w w w w w w w w w w w w w w w w w w w w w w w w w w w w w w w w y y w w w w w w w w w w w w w w w w w w y 0 0 0 0 0 0
Stall 5 (TSP) 0 0 0 0 0 0 0 0 0 w w w w w w w w w w w w w w w w w w y w w w w w w w w w w w w w w w w w w w w w w w w w w w w w w w w w w w w w w w w w w w w w w w w w w w w w w 0 0 0 0 0 0 0 0
Granulation 1SSPA0MAPMAPMAP0 0 0 0 0 0MAPMAPMAP0 0SSP0 0SSPA0 0MAPMAPMAPMAP0 0TSP0MAPMAP0 0 0SSPA0MAPMAPMAPMAPMAPMAP0MAP0 0TSP0MAP0 0SSP0MAPMAPMAPMAPMAP0MAPMAP0 0TSP0MAPMAPMAPMAP0 0SSPA0 0MAPMAPMAPMAPMAP0 0TSPTSPTSP0SSPASSPASSPA0
Granulation 2TSP0 0 0 0 0MAPMAP0MAP0MAP0 0 0SSPA0 0SSPSSP0 0 0TSP0 0TSPTSP0 0 0SSP0MAP0 0 0SSPA0 0 0TSP0 0TSPTSPTSP0MAP0 0 0SSPA0SSP0 0 0 0SSP0TSPTSPTSPTSPTSPTSPTSP0 0SSPSSP0 0SSP0MAPMAPMAPMAPMAPMAPMAP0MAPMAP0MAPMAPMAP
Granulation 3TSP0 0 0 0 0MAPMAP0 0 0MAP0 0 0 0SSP0 0MAPMAP0 0TSP0TSPTSPTSP0 0MAPMAP0 0 0SSPASSPASSPA0 0 0TSP0 0TSPTSPTSP0MAPMAP0 0 0SSPSSPSSP0 0 0MAPMAPMAPMAPMAPMAPMAPMAPMAPMAP0 0SSP0MAP0 0 0 0MAPMAPMAPMAP0 0TSPTSPTSP0MAPMAP
TSP (Granulation) SSP (Granulation) Storage (Stall) H3PO4-MAP (Concentration) SSP (Acidulation)
MAP (Granulation) SSPA (Granulation) H3PO4-TSP (Concentration) TSP (Acidulation)
Loading (Stall)
Curing (Stall)
Fonseca et al (2018)
16
Figure 6. Task allocation in warehouse stalls the scheduling horizon: general model.
0
200
400
600
800
1 10 19 28 37 46 55 64 73 82
Warehouse 1 - Stall 1 (tons)
Loading Curing Storage
0
200
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600
800
1 10 19 28 37 46 55 64 73 82
Warehouse 2 - Stall 1 (tons)
Loading Curing Storage
0
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1 10 19 28 37 46 55 64 73 82
Warehouse 1 - Stall 2 (tons)
Loading Curing Storage
0
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1 10 19 28 37 46 55 64 73 82
Warehouse 2 - Stall 2 (tons)
Loading Curing Storage
0
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1 10 19 28 37 46 55 64 73 82
Warehouse 1 - Stall 3 (tons)
Loading Curing Storage
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1 10 19 28 37 46 55 64 73 82
Warehouse 2 - Stall 3 (tons)
Loading Curing Storage
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1 10 19 28 37 46 55 64 73 82
Warehouse 1 - Stall 4 (tons)
Loading Curing Storage
0
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1 10 19 28 37 46 55 64 73 82
Warehouse 2 - Stall 4 (tons)
Loading Curing Storage
0
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1 10 19 28 37 46 55 64 73 82
Warehouse 1 - Stall 5 (tons)
Loading Curing Storage
0
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1 10 19 28 37 46 55 64 73 82
Warehouse 2 - Stall 5 (tons)
Loading Curing Storage
Journal of Applied Operational Research Vol. 10, No. 1
17
Figure 7. Raw materials consumption profile – general model.
Figure 8. Key continuous units processing rate – general model.
0
20000
40000
60000
80000
100000
120000
1 6 11 16 21 26 31 36 41 46 51 56 61 66 71 76 81 86
Cu
mu
lati
ve R
aw M
ate
rial
Co
nsu
mp
tio
n (
ton
)
Time Periods
Fine Conc.
Gross Conc.
Dry Conc.
H2O
Sulfur
NH3
0
200
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1000
1200
1400
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1 6 11 16 21 26 31 36 41 46 51 56 61 66 71 76 81 86
Pro
cess
ing
Rat
e (t
on
/Per
iod
)
Time Period
H2SO4 - Plant
H3PO4 - Reactor
Acidulation Unit
Fonseca et al (2018)
18
Figure 9. Intermediate materials inventory profiles – general model.
Figure 10. Final product profiles – general model.
0
1000
2000
3000
4000
5000
6000
1 6 11 16 21 26 31 36 41 46 51 56 61 66 71 76 81 86
Inte
rmed
iate
s In
ven
tori
es (
ton
)
Time Periods
H2SO4
H3PO4-TSP
H3PO4-MAP
0
10000
20000
30000
40000
50000
60000
70000
1 6 11 16 21 26 31 36 41 46 51 56 61 66 71 76 81 86
Cu
mu
lati
ve P
rod
uct
ion
(to
n)
Time Periods
SSP
SSPA
TSP
MAP
Journal of Applied Operational Research Vol. 10, No. 1
19
Figure 11. Gantt chart - due date model.
Figure 12. Gantt chart: no production exchanges on weekends.
Figure 13. Gantt chart: general model (4-hour time period).
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90
H2SO4 ProductionH2SO4H2SO4H2SO4H2SO4H2SO4H2SO4H2SO4H2SO4H2SO4H2SO4H2SO4H2SO4H2SO4H2SO4H2SO4H2SO4H2SO4H2SO4H2SO4H2SO4H2SO4H2SO4H2SO4H2SO4H2SO4H2SO4H2SO4H2SO4H2SO4H2SO4H2SO4H2SO4H2SO4H2SO4H2SO4H2SO4H2SO4H2SO4H2SO4H2SO4H2SO4H2SO4H2SO4H2SO4H2SO4H2SO4H2SO4H2SO4H2SO4H2SO4H2SO4H2SO4H2SO4H2SO4H2SO4H2SO4H2SO4H2SO4H2SO4H2SO4H2SO4H2SO4H2SO4H2SO4H2SO4H2SO4H2SO4H2SO4H2SO4H2SO4H2SO4H2SO4H2SO4H2SO4H2SO4H2SO4H2SO4H2SO4H2SO4H2SO4H2SO4H2SO4H2SO4H2SO4H2SO4H2SO4H2SO4H2SO4H2SO4H2SO4
H3PO4 ProductionH3PO4DilH3PO4DilH3PO4DilH3PO4DilH3PO4DilH3PO4DilH3PO4DilH3PO4DilH3PO4DilH3PO4DilH3PO4DilH3PO4DilH3PO4DilH3PO4DilH3PO4DilH3PO4DilH3PO4DilH3PO4DilH3PO4DilH3PO4DilH3PO4DilH3PO4DilH3PO4DilH3PO4DilH3PO4DilH3PO4DilH3PO4DilH3PO4DilH3PO4DilH3PO4DilH3PO4DilH3PO4DilH3PO4DilH3PO4DilH3PO4DilH3PO4DilH3PO4DilH3PO4DilH3PO4DilH3PO4DilH3PO4DilH3PO4DilH3PO4DilH3PO4DilH3PO4DilH3PO4DilH3PO4DilH3PO4DilH3PO4DilH3PO4DilH3PO4DilH3PO4DilH3PO4DilH3PO4DilH3PO4DilH3PO4DilH3PO4DilH3PO4DilH3PO4DilH3PO4DilH3PO4DilH3PO4DilH3PO4DilH3PO4DilH3PO4DilH3PO4DilH3PO4DilH3PO4DilH3PO4DilH3PO4DilH3PO4DilH3PO4DilH3PO4DilH3PO4DilH3PO4DilH3PO4DilH3PO4DilH3PO4DilH3PO4DilH3PO4DilH3PO4DilH3PO4DilH3PO4DilH3PO4DilH3PO4DilH3PO4DilH3PO4DilH3PO4DilH3PO4DilH3PO4Dil
H3PO4 ConcentrationH3PO4-MAPH3PO4-TSPH3PO4-MAPH3PO4-MAPH3PO4-MAPH3PO4-TSPH3PO4-TSPH3PO4-MAPH3PO4-TSPH3PO4-MAPH3PO4-MAPH3PO4-MAPH3PO4-MAPH3PO4-TSPH3PO4-MAPH3PO4-TSPH3PO4-MAPH3PO4-MAPH3PO4-TSPH3PO4-TSPH3PO4-TSPH3PO4-MAPH3PO4-MAPH3PO4-MAPH3PO4-MAPH3PO4-MAPH3PO4-MAPH3PO4-TSPH3PO4-MAPH3PO4-MAPH3PO4-TSPH3PO4-MAPH3PO4-MAPH3PO4-MAPH3PO4-MAPH3PO4-MAPH3PO4-MAPH3PO4-MAPH3PO4-TSPH3PO4-MAPH3PO4-MAPH3PO4-MAPH3PO4-TSPH3PO4-MAPH3PO4-MAPH3PO4-TSPH3PO4-MAPH3PO4-MAPH3PO4-TSPH3PO4-MAPH3PO4-TSPH3PO4-MAPH3PO4-MAPH3PO4-MAPH3PO4-TSPH3PO4-MAPH3PO4-MAPH3PO4-MAPH3PO4-MAPH3PO4-MAPH3PO4-MAPH3PO4-MAPH3PO4-MAPH3PO4-MAPH3PO4-MAPH3PO4-MAPH3PO4-MAPH3PO4-MAPH3PO4-MAPH3PO4-MAPH3PO4-MAPH3PO4-MAPH3PO4-MAPH3PO4-MAPH3PO4-MAPH3PO4-MAPH3PO4-MAPH3PO4-MAPH3PO4-MAPH3PO4-MAPH3PO4-MAPH3PO4-MAPH3PO4-MAPH3PO4-MAPH3PO4-MAPH3PO4-MAPH3PO4-MAPH3PO4-MAPH3PO4-MAPH3PO4-MAP
AcidulationSSPSSPSSPSSP0TSPTSPTSPTSPTSP0 0 0 0 0 0SSPSSPSSPSSP0SSP0TSPTSPTSPTSP0TSP0 0 0 0SSP0SSP0SSPSSP0 0TSPTSP0TSPTSP0TSP0 0SSP0SSPSSPSSP0 0 0 0 0 0TSPTSPTSPTSP0 0TSP0SSPSSP0SSP0SSP0 0 0 0 0 0 0 0 0 0 0 0 0 0 0
Stall 1 (SSP) 0 w w w w w w w w w w w w w w w y x w w w w w w w w w w w w w w w y 0 0 0 w w w w w w w w w w w w w w w x w w w w w w w w w w w w w w w y 0 x w w w w w w w w w w w w w w w 0 0 0 0
Stall 2 (SSP) x x w w w w w w w w w w w w w w w w w w w w w w w w w w w w w w y 0 0 w w w w w w w w w w w w w w w w w w w w w w w w w w w w w w y y y 0 x x x w w w w w w w w w w w w w w w y 0 0
Stall 3 (SSP) w w w w w w w w w w w w w w w y x x w w w w w w w w w w w w w w w w w w w w w w w w w w w w w w y y y y x w w w w w w w w w w w w w w w 0 w w w w w w w w w w w w w w w y y 0 0 0 0
Stall 4 (SSP) 0 0 0 w w w w w w w w w w w w w w w y w w w w w w w w w w w w w w w 0 x x x w w w w w w w w w w w w w w w y w w w w w w w w w w w w w w w y y 0 0 0 w w w w w w w w w w w w w w w 0
Stall 5 (SSP) 0 0 x w w w w w w w w w w w w w w w y 0 0 w w w w w w w w w w w w w w w 0 x w w w w w w w w w w w w w w w x w w w w w w w w w w w w w w w y y 0 w w w w w w w w w w w w w w w y y 0
Stall 1 (TSP) y y y 0 0 0 0 w w w w w w w w w w w w w w w w w w w w w w w w w w w w w w w w w w w w y w w w w w w w w w w w w w w w w w w w w w w w w w w w w w w w w w w w w y y y y y y 0 0 0 0
Stall 2 (TSP) 0 0 0 0 0 0 0 0 x w w w w w w w w w w w w w w w w w w 0 w w w w w w w w w w w w w w w w w w 0 w w w w w w w w w w w w w w w w w w y 0 x w w w w w w w w w w w w w w w w w w 0 0 0 0
Stall 3 (TSP) 0 0 0 0 0 0 w w w w w w w w w w w w w w w w w w w w w w w w w w w w w w w w w w w w w w w w w w w w w w w w w w w w w w y 0 0 0 w w w w w w w w w w w w w w w w w w y 0 0 0 0 0 0 0
Stall 4 (TSP) 0 0 0 0 0 w w w w w w w w w w w w w w w w w w w w w w w w w w w w w w w w w w w w w w w w w w w w w w w w w w w w w w y 0 w w w w w w w w w w w w w w w w w w y y 0 0 0 0 0 0 0 0 0
Stall 5 (TSP) 0 0 0 0 0 0 x x w w w w w w w w w w w w w w w w w w w w w w w w w w w w w w w w w w w w 0 w w w w w w w w w w w w w w w w w w w w w w w w w w w w w w w w w w w w y 0 0 0 0 0 0 0 0
Granulation 1SSPA0TSPTSP0MAP0MAP0MAPMAP0MAP0 0 0SSPSSP0SSPA0MAPMAPMAPMAPMAPMAP0MAPMAPMAP0 0SSPA0MAPMAP0 0MAPMAP0MAPMAP0 0TSP0MAPMAP0 0MAPMAPMAPMAPMAPMAPMAPMAPMAPMAPMAPMAPMAPMAP0MAPMAPMAPMAPMAPMAPMAPMAPMAPMAP0MAP0MAPMAPMAP0 0 0TSP0SSPA0
Granulation 2 0 0MAPMAP0 0 0MAP0MAPMAP0MAP0 0 0 0SSPA0 0MAP0 0TSPTSPTSPTSPTSP0 0MAP0 0SSPASSPA0SSP0MAP0 0TSPTSPTSPTSP0TSP0 0 0SSPA0SSPASSPA0MAP0 0 0TSPTSPTSPTSPTSP0TSPTSP0SSPASSPA0SSPA0 0MAP0MAP0 0TSPTSPTSPTSPTSP0SSPSSP0MAPMAP
Granulation 3 0MAP0 0 0 0 0 0 0MAP0MAP0 0 0 0SSP0 0SSPA0 0 0TSPTSPTSPTSPTSP0 0MAP0 0SSPSSP0 0 0MAP0 0TSPTSP0TSP0MAPMAP0 0SSPSSPSSPSSPSSP0MAPMAPMAPMAP0 0TSPTSP0 0SSPA0SSPSSP0SSP0 0 0 0MAPMAPMAPMAPMAPMAP0MAP0MAP0MAPMAPMAP
TSP (Granulation) SSP (Granulation) Storage (Stall) H3PO4-MAP (Concentration) SSP (Acidulation)
MAP (Granulation) SSPA (Granulation) H3PO4-TSP (Concentration) TSP (Acidulation)
Loading (Stall)
Curing (Stall)
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H2SO4 ProductionH2SO4H2SO4H2SO4H2SO4H2SO4H2SO4H2SO4H2SO4H2SO4H2SO4H2SO4H2SO4H2SO4H2SO4H2SO4H2SO4H2SO4H2SO4H2SO4H2SO4H2SO4H2SO4H2SO4H2SO4H2SO4H2SO4H2SO4H2SO4H2SO4H2SO4H2SO4H2SO4H2SO4H2SO4H2SO4H2SO4H2SO4H2SO4H2SO4H2SO4H2SO4H2SO4H2SO4H2SO4H2SO4H2SO4H2SO4H2SO4H2SO4H2SO4H2SO4H2SO4H2SO4H2SO4H2SO4H2SO4H2SO4H2SO4H2SO4H2SO4H2SO4H2SO4H2SO4H2SO4H2SO4H2SO4H2SO4H2SO4H2SO4H2SO4H2SO4H2SO4H2SO4H2SO4H2SO4H2SO4H2SO4H2SO4H2SO4H2SO4H2SO4H2SO4H2SO4H2SO4H2SO4H2SO4H2SO4H2SO4H2SO4H2SO4
H3PO4 ProductionH3PO4DilH3PO4DilH3PO4DilH3PO4DilH3PO4DilH3PO4DilH3PO4DilH3PO4DilH3PO4DilH3PO4DilH3PO4DilH3PO4DilH3PO4DilH3PO4DilH3PO4DilH3PO4DilH3PO4DilH3PO4DilH3PO4DilH3PO4DilH3PO4DilH3PO4DilH3PO4DilH3PO4DilH3PO4DilH3PO4DilH3PO4DilH3PO4DilH3PO4DilH3PO4DilH3PO4DilH3PO4DilH3PO4DilH3PO4DilH3PO4DilH3PO4DilH3PO4DilH3PO4DilH3PO4DilH3PO4DilH3PO4DilH3PO4DilH3PO4DilH3PO4DilH3PO4DilH3PO4DilH3PO4DilH3PO4DilH3PO4DilH3PO4DilH3PO4DilH3PO4DilH3PO4DilH3PO4DilH3PO4DilH3PO4DilH3PO4DilH3PO4DilH3PO4DilH3PO4DilH3PO4DilH3PO4DilH3PO4DilH3PO4DilH3PO4DilH3PO4DilH3PO4DilH3PO4DilH3PO4DilH3PO4DilH3PO4DilH3PO4DilH3PO4DilH3PO4DilH3PO4DilH3PO4DilH3PO4DilH3PO4DilH3PO4DilH3PO4DilH3PO4DilH3PO4DilH3PO4DilH3PO4DilH3PO4DilH3PO4DilH3PO4DilH3PO4DilH3PO4DilH3PO4Dil
H3PO4 ConcentrationH3PO4-TSPH3PO4-TSPH3PO4-MAPH3PO4-MAPH3PO4-TSPH3PO4-TSPH3PO4-TSPH3PO4-MAPH3PO4-MAPH3PO4-MAPH3PO4-MAPH3PO4-MAPH3PO4-MAPH3PO4-TSPH3PO4-MAPH3PO4-MAPH3PO4-MAPH3PO4-MAPH3PO4-MAPH3PO4-MAPH3PO4-MAPH3PO4-MAPH3PO4-MAPH3PO4-TSPH3PO4-TSPH3PO4-MAPH3PO4-MAPH3PO4-TSPH3PO4-MAPH3PO4-MAPH3PO4-MAPH3PO4-MAPH3PO4-MAPH3PO4-MAPH3PO4-MAPH3PO4-TSPH3PO4-MAPH3PO4-MAPH3PO4-MAPH3PO4-MAPH3PO4-MAPH3PO4-MAPH3PO4-MAPH3PO4-TSPH3PO4-MAPH3PO4-MAPH3PO4-MAPH3PO4-TSPH3PO4-TSPH3PO4-MAPH3PO4-MAPH3PO4-MAPH3PO4-MAPH3PO4-MAPH3PO4-MAPH3PO4-TSPH3PO4-MAPH3PO4-MAPH3PO4-MAPH3PO4-MAPH3PO4-MAPH3PO4-MAPH3PO4-MAPH3PO4-MAPH3PO4-TSPH3PO4-TSPH3PO4-MAPH3PO4-TSPH3PO4-MAPH3PO4-MAPH3PO4-MAPH3PO4-MAPH3PO4-MAPH3PO4-MAPH3PO4-MAPH3PO4-MAPH3PO4-MAPH3PO4-MAPH3PO4-MAPH3PO4-MAPH3PO4-MAPH3PO4-MAPH3PO4-MAPH3PO4-MAPH3PO4-MAPH3PO4-MAPH3PO4-MAPH3PO4-MAPH3PO4-MAPH3PO4-MAP
AcidulationSSPSSPSSPSSP0TSPTSPTSPTSP0TSP0 0 0 0SSP0SSPSSPSSPSSP0 0 0TSPTSPTSPTSP0 0 0TSP0 0 0SSPSSPSSPSSP0 0 0 0 0 0TSPTSPTSPTSPTSP0 0SSPSSP0SSP0SSP0 0 0 0 0 0TSPTSPTSPTSPTSP0 0SSPSSPSSPSSP0 0 0 0 0 0 0 0 0 0 0 0 0 0 0
Stall 1 (SSP) y 0 w w w w w w w w w w w w w w w w w w w w w w w w w w w w w w y y 0 0 0 w w w w w w w w w w w w w w w w w w w w w w w w w w w w w w y y 0 0 x x w w w w w w w w w w w w w w w y 0
Stall 2 (SSP) w w w w w w w w w w w w w w w x x x w w w w w w w w w w w w w w w y y x x w w w w w w w w w w w w w w w y y 0 x w w w w w w w w w w w w w w w 0 x w w w w w w w w w w w w w w w 0 0
Stall 3 (SSP) x x w w w w w w w w w w w w w w w y y x w w w w w w w w w w w w w w w x w w w w w w w w w w w w w w w 0 0 w w w w w w w w w w w w w w w y y y w w w w w w w w w w w w w w w y y 0 0
Stall 4 (SSP) 0 w w w w w w w w w w w w w w w 0 0 w w w w w w w w w w w w w w w y 0 0 w w w w w w w w w w w w w w w y x x x x x w w w w w w w w w w w w w w w y y w w w w w w w w w w w w w w w 0
Stall 5 (SSP) 0 0 x w w w w w w w w w w w w w w w x w w w w w w w w w w w w w w w 0 0 0 0 w w w w w w w w w w w w w w w y 0 0 0 w w w w w w w w w w w w w w w w w w w w w w w w w w w w w w y 0 0
Stall 1 (TSP) y y y y y 0 0 w w w w w w w w w w w w w w w w w w 0 w w w w w w w w w w w w w w w w w w y y 0 x w w w w w w w w w w w w w w w w w w w w w w w w w w w w w w w w w w w w 0 0 0 0 0 0
Stall 2 (TSP) 0 0 0 0 0 0 0 0 w w w w w w w w w w w w w w w w w w 0 w w w w w w w w w w w w w w w w w w w w w w w w w w w w w w w w w w w w y y w w w w w w w w w w w w w w w w w w 0 0 0 0 0 0 0
Stall 3 (TSP) 0 0 0 0 0 0 0 0 0 0 w w w w w w w w w w w w w w w w w w 0 0 0 w w w w w w w w w w w w w w w w w w x w w w w w w w w w w w w w w w w w w w w w w w w w w w w w w w w w w w w 0 0 0 0
Stall 4 (TSP) 0 0 0 0 0 0 w w w w w w w w w w w w w w w w w w y w w w w w w w w w w w w w w w w w w y 0 x w w w w w w w w w w w w w w w w w w w w w w w w w w w w w w w w w w w w 0 0 0 0 0 0 0 0
Stall 5 (TSP) 0 0 0 0 0 x w w w w w w w w w w w w w w w w w w x w w w w w w w w w w w w w w w w w w 0 0 0 0 0 w w w w w w w w w w w w w w w w w w y w w w w w w w w w w w w w w w w w w 0 0 0 0 0
Granulation 1 0SSP0 0MAPMAPMAP0MAPMAPMAPMAPMAP0 0MAP0MAP0MAP0 0 0 0TSPTSPTSP0TSP0 0 0SSPA0SSPASSPA0 0 0 0 0 0 0TSPTSPTSP0 0 0TSP0SSPSSPSSP0 0MAPMAPMAPMAPMAPMAP0 0TSPTSP0 0TSP0 0 0 0 0 0 0 0 0 0 0 0 0TSPTSPTSPTSPTSP0SSPA0
Granulation 2 0 0 0 0 0TSP0 0 0 0MAP0MAP0 0SSPSSPSSPSSPSSP0 0MAP0MAPMAP0 0MAPMAPMAP0MAPMAPMAP0 0 0MAPMAPMAPMAP0 0TSPTSPTSP0 0 0 0 0SSP0MAPMAP0MAPMAPMAPMAPMAPMAP0 0 0TSP0 0SSP0SSPASSPA0 0MAPMAPMAPMAP0MAPMAPMAPMAPMAP0 0SSPASSPASSPA
Granulation 3 0 0 0 0 0TSP0 0 0 0 0 0 0 0 0MAP0 0MAP0MAPMAP0 0TSPTSPTSP0TSP0MAP0 0SSPASSPA0MAPMAP0MAP0MAPMAP0MAPMAP0 0MAP0 0SSPA0 0SSP0MAPMAPMAPMAPMAPMAPMAPMAP0 0TSPTSPTSP0SSPASSPASSPA0SSPA0MAP0MAPMAPMAPMAPMAPMAPMAPMAPMAPMAPMAPMAP
TSP (Granulation) SSP (Granulation) Storage (Stall) H3PO4-MAP (Concentration) SSP (Acidulation)
MAP (Granulation) SSPA (Granulation) H3PO4-TSP (Concentration) TSP (Acidulation)
Loading (Stall)
Curing (Stall)
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H2SO4 ProductionH2SO4H2SO4H2SO4H2SO4H2SO4H2SO4H2SO4H2SO4H2SO4H2SO4H2SO4H2SO4H2SO4H2SO4H2SO4H2SO4H2SO4H2SO4H2SO4H2SO4H2SO4H2SO4H2SO4H2SO4H2SO4H2SO4H2SO4H2SO4H2SO4H2SO4H2SO4H2SO4H2SO4H2SO4H2SO4H2SO4H2SO4H2SO4H2SO4H2SO4H2SO4H2SO4H2SO4H2SO4H2SO4H2SO4H2SO4H2SO4H2SO4H2SO4H2SO4H2SO4H2SO4H2SO4H2SO4H2SO4H2SO4H2SO4H2SO4H2SO4H2SO4H2SO4H2SO4H2SO4H2SO4H2SO4H2SO4H2SO4H2SO4H2SO4H2SO4H2SO4H2SO4H2SO4H2SO4H2SO4H2SO4H2SO4H2SO4H2SO4H2SO4H2SO4H2SO4H2SO4H2SO4H2SO4H2SO4H2SO4H2SO4H2SO4H2SO4H2SO4H2SO4H2SO4H2SO4H2SO4H2SO4H2SO4H2SO4H2SO4H2SO4H2SO4H2SO4H2SO4H2SO4H2SO4H2SO4H2SO4H2SO4H2SO4H2SO4H2SO4H2SO4H2SO4H2SO4H2SO4H2SO4H2SO4H2SO4H2SO4H2SO4H2SO4H2SO4H2SO4H2SO4H2SO4H2SO4H2SO4H2SO4H2SO4H2SO4H2SO4H2SO4H2SO4H2SO4H2SO4H2SO4H2SO4H2SO4H2SO4H2SO4H2SO4H2SO4H2SO4H2SO4H2SO4H2SO4H2SO4H2SO4H2SO4H2SO4H2SO4H2SO4H2SO4H2SO4H2SO4H2SO4H2SO4H2SO4H2SO4H2SO4H2SO4H2SO4H2SO4H2SO4H2SO4H2SO4H2SO4H2SO4H2SO4H2SO4H2SO4H2SO4H2SO4H2SO4H2SO4H2SO4H2SO4H2SO4H2SO4
H3PO4 ProductionH3PO4DilH3PO4DilH3PO4DilH3PO4DilH3PO4DilH3PO4DilH3PO4DilH3PO4DilH3PO4DilH3PO4DilH3PO4DilH3PO4DilH3PO4DilH3PO4DilH3PO4DilH3PO4DilH3PO4DilH3PO4DilH3PO4DilH3PO4DilH3PO4DilH3PO4DilH3PO4DilH3PO4DilH3PO4DilH3PO4DilH3PO4DilH3PO4DilH3PO4DilH3PO4DilH3PO4DilH3PO4DilH3PO4DilH3PO4DilH3PO4DilH3PO4DilH3PO4DilH3PO4DilH3PO4DilH3PO4DilH3PO4DilH3PO4DilH3PO4DilH3PO4DilH3PO4DilH3PO4DilH3PO4DilH3PO4DilH3PO4DilH3PO4DilH3PO4DilH3PO4DilH3PO4DilH3PO4DilH3PO4DilH3PO4DilH3PO4DilH3PO4DilH3PO4DilH3PO4DilH3PO4DilH3PO4DilH3PO4DilH3PO4DilH3PO4DilH3PO4DilH3PO4DilH3PO4DilH3PO4DilH3PO4DilH3PO4DilH3PO4DilH3PO4DilH3PO4DilH3PO4DilH3PO4DilH3PO4DilH3PO4DilH3PO4DilH3PO4DilH3PO4DilH3PO4DilH3PO4DilH3PO4DilH3PO4DilH3PO4DilH3PO4DilH3PO4DilH3PO4DilH3PO4DilH3PO4DilH3PO4DilH3PO4DilH3PO4DilH3PO4DilH3PO4DilH3PO4DilH3PO4DilH3PO4DilH3PO4DilH3PO4DilH3PO4DilH3PO4DilH3PO4DilH3PO4DilH3PO4DilH3PO4DilH3PO4DilH3PO4DilH3PO4DilH3PO4DilH3PO4DilH3PO4DilH3PO4DilH3PO4DilH3PO4DilH3PO4DilH3PO4DilH3PO4DilH3PO4DilH3PO4DilH3PO4DilH3PO4DilH3PO4DilH3PO4DilH3PO4DilH3PO4DilH3PO4DilH3PO4DilH3PO4DilH3PO4DilH3PO4DilH3PO4DilH3PO4DilH3PO4DilH3PO4DilH3PO4DilH3PO4DilH3PO4DilH3PO4DilH3PO4DilH3PO4DilH3PO4DilH3PO4DilH3PO4DilH3PO4DilH3PO4DilH3PO4DilH3PO4DilH3PO4DilH3PO4DilH3PO4DilH3PO4DilH3PO4DilH3PO4DilH3PO4DilH3PO4DilH3PO4DilH3PO4DilH3PO4DilH3PO4DilH3PO4DilH3PO4DilH3PO4DilH3PO4DilH3PO4DilH3PO4DilH3PO4DilH3PO4DilH3PO4DilH3PO4DilH3PO4DilH3PO4DilH3PO4DilH3PO4DilH3PO4DilH3PO4DilH3PO4DilH3PO4DilH3PO4Dil
H3PO4 ConcentrationH3PO4-MAPH3PO4-MAPH3PO4-TSPH3PO4-MAPH3PO4-MAPH3PO4-MAPH3PO4-MAPH3PO4-MAPH3PO4-MAPH3PO4-MAPH3PO4-MAPH3PO4-TSPH3PO4-TSPH3PO4-MAPH3PO4-MAPH3PO4-TSPH3PO4-TSPH3PO4-MAPH3PO4-TSPH3PO4-TSPH3PO4-MAPH3PO4-MAPH3PO4-MAPH3PO4-MAPH3PO4-MAPH3PO4-MAPH3PO4-MAPH3PO4-MAPH3PO4-MAPH3PO4-MAPH3PO4-MAPH3PO4-MAPH3PO4-MAPH3PO4-MAPH3PO4-MAPH3PO4-MAPH3PO4-TSPH3PO4-MAPH3PO4-MAPH3PO4-TSPH3PO4-MAPH3PO4-MAPH3PO4-MAPH3PO4-MAPH3PO4-TSPH3PO4-MAPH3PO4-MAPH3PO4-TSPH3PO4-MAPH3PO4-MAPH3PO4-TSPH3PO4-TSPH3PO4-MAPH3PO4-MAPH3PO4-MAPH3PO4-MAPH3PO4-TSPH3PO4-MAPH3PO4-MAPH3PO4-TSPH3PO4-MAPH3PO4-TSPH3PO4-MAPH3PO4-MAPH3PO4-TSPH3PO4-TSPH3PO4-TSPH3PO4-MAPH3PO4-MAPH3PO4-MAPH3PO4-MAPH3PO4-MAPH3PO4-MAPH3PO4-MAPH3PO4-TSPH3PO4-MAPH3PO4-MAPH3PO4-MAPH3PO4-MAPH3PO4-TSPH3PO4-MAPH3PO4-MAPH3PO4-MAPH3PO4-MAPH3PO4-MAPH3PO4-MAPH3PO4-TSPH3PO4-MAPH3PO4-MAPH3PO4-MAPH3PO4-MAPH3PO4-MAPH3PO4-MAPH3PO4-MAPH3PO4-MAPH3PO4-MAPH3PO4-TSPH3PO4-TSPH3PO4-MAPH3PO4-MAPH3PO4-MAPH3PO4-MAPH3PO4-MAPH3PO4-MAPH3PO4-MAPH3PO4-MAPH3PO4-MAPH3PO4-MAPH3PO4-MAPH3PO4-MAPH3PO4-MAPH3PO4-MAPH3PO4-MAPH3PO4-MAPH3PO4-MAPH3PO4-MAPH3PO4-MAPH3PO4-MAPH3PO4-TSPH3PO4-MAPH3PO4-MAPH3PO4-MAPH3PO4-MAPH3PO4-MAPH3PO4-MAPH3PO4-TSPH3PO4-MAPH3PO4-TSPH3PO4-MAPH3PO4-MAPH3PO4-MAPH3PO4-TSPH3PO4-TSPH3PO4-TSPH3PO4-TSPH3PO4-MAPH3PO4-MAPH3PO4-MAPH3PO4-MAPH3PO4-MAPH3PO4-MAPH3PO4-MAPH3PO4-MAPH3PO4-MAPH3PO4-MAPH3PO4-MAPH3PO4-MAPH3PO4-MAPH3PO4-MAPH3PO4-MAPH3PO4-MAPH3PO4-MAPH3PO4-MAPH3PO4-MAPH3PO4-MAPH3PO4-MAPH3PO4-MAPH3PO4-MAPH3PO4-MAPH3PO4-MAPH3PO4-MAPH3PO4-MAPH3PO4-MAPH3PO4-MAPH3PO4-MAPH3PO4-MAPH3PO4-MAPH3PO4-MAPH3PO4-MAPH3PO4-MAPH3PO4-MAPH3PO4-MAPH3PO4-MAPH3PO4-MAPH3PO4-MAPH3PO4-MAPH3PO4-MAPH3PO4-MAPH3PO4-MAPH3PO4-MAP
AcidulationSSPSSPSSPSSPSSPSSPSSPSSP TSPTSPTSPTSPTSPTSPTSPTSPTSPTSP SSPSSPSSPSSPSSPSSPSSPSSPSSP TSPTSPTSPTSPTSPTSPTSPTSPTSPTSP SSPSSPSSPSSPSSPSSPSSPSSPSSPSSP TSPTSPTSPTSPTSPTSPTSPTSPTSPTSPSSPSSP SSPSSPSSPSSPSSPSSPSSPSSP TSPTSPTSPTSPTSPTSPTSPTSPTSPTSPSSPSSP SSPSSPSSPSSPSSPSSP
Stall 1 (SSP) xwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwy xwwwwwwwwwwwwwwwwwwwwwwwwwwwwww xwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwxxwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwyyyyyyxwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwyyyyy
Stall 2 (SSP) xwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwy xwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwyyyy xxwwwwwwwwwwwwwwwwwwwwwwwwwwwxxwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwyyyyyyxwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwyyyyy
Stall 3 (SSP) xxxxxxxwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwyyyyxwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwyxxxxwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwyyyyxwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwyyxwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwyy
Stall 4 (SSP) xxxxxxxxxxxxxwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwxwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwyyy xxwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwyxwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwxwwwwwwwwwwwwwwwwwwwwwwwwwwwwww
Stall 5 (SSP) xxxwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwy xwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwyyyyyy xxxxwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwyxxwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwyyxwwwwwwwwwwwwwwwwwwwwwwwwwwwwww
Stall 1 (TSP) yyyy xxwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwyyy xxwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwxwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwyxwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwy
Stall 2 (TSP) xwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwxwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwyyxwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwyxxxwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwy
Stall 3 (TSP) xxxwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwww xxwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwyxxxwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwxxxwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwyy
Stall 4 (TSP) xwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwyxwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwww xwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwxwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwy
Stall 5 (TSP) xwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwww xxxwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwxwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwyxwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwy
Granulation 1SSPAMAPMAPMAPMAPMAPMAPMAPMAPMAPMAPMAPMAPMAPMAPMAPMAPMAP SSPASSPASSPASSPAMAP SSPA TSPTSPTSPTSPTSP MAPMAPMAP SSPSSPSSPASSPASSPASSPAMAP TSPTSPTSPTSPTSPTSPTSPTSPTSPTSPSSPSSP SSPSSPSSPSSPSSPSSPSSPSSP TSPTSPTSPTSPTSPTSPTSPTSPTSPTSPSSPSSP SSPSSPSSPSSPSSPSSP
Granulation 2SSP MAPMAPMAPMAP MAPMAPMAPMAPMAPMAPMAP SSPASSPASSPA TSPTSP TSP MAPMAPMAPMAP SSPSSP MAPMAPMAPMAPMAPMAPMAP TSPTSPTSPTSPTSPTSPTSPMAPMAPMAPMAPMAPMAPMAPMAPMAPMAPMAPMAPMAPMAPMAPMAPMAPMAP TSPTSPTSPTSPTSPTSPMAPMAP SSPSSP MAPMAPMAPMAPMAPMAPMAPMAPMAPMAPMAPMAPMAPMAPMAPMAPMAPMAPMAPMAPMAPMAPMAPMAPMAPMAP
Granulation 3 TSPTSPTSP MAPMAP MAPMAPMAPMAP SSPASSPASSPA TSPTSP TSPTSPTSPMAPMAPMAP SSPAMAPMAPMAPMAPMAPMAP TSPTSPTSPTSP SSPMAPMAPMAPMAPMAPMAPMAPMAPMAPMAPMAPMAPMAPMAPMAPMAP TSPTSPMAPMAPMAPMAPMAPMAPMAPMAPMAPMAPMAPMAP MAPMAPMAPMAPMAP TSPTSPTSPSSPASSPASSPASSPASSPASSPA
TSP (Granulation) SSP (Granulation) Loading (Stall) Storage (Stall) H3PO4-MAP (Concentration) SSP (Acidulation)
MAP (Granulation) SSPA (Granulation) Curing (Stall) H3PO4-TSP (Concentration) TSP (Acidulation)
Fonseca et al (2018)
20
Table 4. Demand versus production – due date model
Product Demand incident at
the first half of the time
horizon (tons)
Production in first
half of the time horizon
(tons)
Additional demand
incident at the end of the
time horizon (tons)
Production in second
half of the time horizon
(tons)
SSP 2,000 3,200 6,000 4,800
SSPA 3,500 4,500 6,500 5,700
TSP 5,000 9,000 10,000 9.900
MAP 6,500 24,680 12,000 40,320
Table 5. Case studies statistics results*
General model Due date Weekend General model Due date Weekend
(8-hour time interval) (4-hour time interval)
Relaxed MILP ($) 25,896,000 25,896,000 25,896,000 25,896,000 25,896,000 25,896,000
MILP ($) 25,896,000 25,896,000 25,896,000 25,896,000 25,896,000 25,896,000
Equations 15,636 15,640 18,020 31,296 31,300 35,120
Continuous var. 5,852 5,852 9,452 11,702 11,702 18,902
Binary variables 4,320 4,320 5,760 8,640 8,640 11,520
N° of iterations 135,690 119,557 99,217 2,671,300 804,549 2,420,903
Nodes 984 533 502 6,732 1,415 3,863
CPU (s) 52.369 31.964 44.117 1628.229 858.490 3,373.615
Relative gap (%) 0.0 0.0 0.0 0.0 0.0 0.0
*Processor: Intel® Core ™ i7-2860QM CPU @ 2.50GHz (8 GB RAM).
Figure 7 shows raw material consumption. It can be observed an intense use of gross concentrate as a result of its high
consumption factor in H3PO4 recipe. H3PO4 also demands large amounts of H2SO4 leading to a high consumption of
sulfur together with fine concentrate, which is also needed for the acidulation step. The sulfuric acid production plant
operated continuously and at full load (1,200 tons/period) while phosphoric acid production and the acidulation unit
oscillated between their minimum and maximum capacities, Figure 8. Even though H2SO4 production was at full load, its
stock was observed to be at its minimum for most of the time along the scheduling horizon (Figure 9). This unit is thus
identified as the main bottleneck to production increase of any of the final products. No restrictions were added to ensure
maintenance of minimum intermediate stocks at the horizon’s end, which may be a good policy to guarantee the availability
of intermediates for the allocation of subsequent productive resources, early in the ensuing horizon. The profiles of final
product production are shown in Figure 10. It is clear by this figure the massive MAP production given that it does not
depend on the curing step, which enables its production at the first time periods, whereas the other products production is
mainly perceived only after the sixteenth time period.
Table 4 shows the values of intermediate demands and optimized production results for the second case. Due dates
were imposed exactly at half the time horizon although the implementation of smaller periodic intervals would be easy.
Figure 11 shows the corresponding Gantt chart corresponding to this problem. Like with case 1, the acidulation unit as
well as the granulation units showed a significant number of changeovers and a high utilization of both warehouses,
although operations had to be reorganized in order to satisfy the demand before the end of the scheduling horizon.
The proposed model also investigated the effect of an important operational rule through case study 3: the prohibition
of weekend changeover due to lack of supervisory personnel in such periods. During weekends, the real plant upon which
this study is based is under the responsibility of the operators. By prohibiting changes on weekends, accidents causing
property and individual damages are prevented. The Gantt chart in Figure 12 illustrates how scheduled production optimization
accounted for such logistical restrictions. No changeover was scheduled for the time periods contained inside the dashed
lines. Table 2 compares the problem’s objective function, both with and without this restriction’s imposition, proving that
the optimization found an alternative solution satisfying that restriction without impacting the company's profit. Finally, a sensitive analysis was conducted as to give an idea on how computational performance is affected when more
accurate time periods are used for the problem under study. Obviously, time period length must be chosen as the greatest
common factor of all tasks. Therefore, an 8-hour time period was suitable for addressing the problem with the data given
the appendix. Was changeover times a multiple of a 4-hour time period solution time would had increased by a factor of 31
Journal of Applied Operational Research Vol. 10, No. 1
21
for the general model, as a result of the increase in model size, as shown in Table 5. Note that by solving the general
model with a more refined time grid does not change the objective function value, whose justification should be based on
the argument that by increasing the number of points in which tasks can start or finish (beyond the number determined by
the greatest common factor method) does not change the feasible region. One benefit though, was the more detailed operation
as can be seen by the Gantt chart presented in Figure 13. Note that the loading, curing and storage steps at stalls were
more apparent. Moreover, a downside of increasing accuracy was the increased problem dimension which drastically
affected computational performance, especially for the case involving barring changeovers on weekend days.
Conclusions
A mathematical model for a multipurpose phosphate fertilizer plant (incorporating actual relevant operational details) was
proposed on the basis of discrete time representation. The problems used to study the computational performance behavior
of the model were based on a real-world plant information. The scheduling horizon of a fertilizer production plant is typically
higher than most production scheduling problems. One month is the minimum period capable of providing reasonable
schedules, given the long curing times of intermediate products used in this plant type. When using discrete time representation,
the generation of large models is thus expected. For all instances studied, the large size drawback was attenuated by
expanding time horizons as much as possible so as to capture the necessary detail level. In addition to their large dimensions,
studied problems have multiple symmetric solutions (typical of production scheduling problems). However, no attempt
was made to explore the latter aspect. Computational performance results indicate that it is possible to obtain solutions
with reasonable computational times, keeping the modeling simple by adopting discrete time representation. Specific
requirements such as satisfaction of demands distributed along the time horizon (due dates) and the prohibition of
weekend changeover due to lack of supervisory personnel in such periods were introduced without impacting the computational
performance. Important operational aspects were analyzed in the discussion section making clear that the formulation is
suitable for managing a fertilizer production plant. It was also demonstrated that increasing time granularity did not necessarily
cause an improvement in the optimal solution value to the cases addressed in this paper although a more detailed operation
could be achieved. Efforts are being devoted to the development of a model with continuous time representation for comparing
it against the discrete time representation one. This will help establish the pros and cons of each method when addressing
the specific problem posed by the fertilizer industry.
References
Abdul-Wahab, S. A.; Failaka, M. F.; Ahmadi, L., Elkamel, A.; Yetilmezsoy, K. Nonlinear programming optimization of
series and parallel cyclone arrangement of NPK fertilizer plants. Powder Technology, 264, 203-215, 2014.
Barbosa-Povoa, A. P.; Pantelides, C. C. Design of multipurpose plants using the resource-task network unified frame-
work. Computers & Chemical Engineering, 21, S703-S708, 1997.
Castro, P. M.; Barbosa-Povoa, A. P.; Matos, H. A.; Novais, A. Q. Simple continuous-time formulation for short-term
scheduling of batch and continuous processes. Industrial & Engineering Chemistry Research, 43, 105-118, 2004.
Castro, P. M.; Barbosa-Póvoa, A.P.; Matos, H. A. Simple Continuous-Time Formulation for Short-Term scheduling of
Batch and Continuous Processes. Industrial & Engineering Chemistry Research, 43, 105-118, 2004.
Castro, P.; Barbosa-Póvoa, A.P.F.D.; Matos, H.A. An Improved RTN Continuous-Time Formulation for the Short-term
Scheduling of Multipurpose Batch Plants. Industrial & Engineering Chemistry Research, 40, 2059-2068, 2001.
Cekinski, E.; Calmonovici, C. E.; Bichara, J. M.; Fabiani, M.; Giulietti, M.; Castro, M. L. M. M.; Silveira, P. B. M.;
Pressinotti, Q. S. H. C.; Guardani, R. (Ed.). Tecnologia de produção de fertilizantes. São Paulo: Instituto de Pesquisas
Tecnológicas, 1990. p. 95-129.
Dawson, C. J.; Hilton, J. Fertilizer availability in a resource- limited world: Production and recycling of nitrogen and
phosphorus. Food Policy, 36, S14-S22, 2011.
Floudas, C. A.; Lin, X. Continuous-time Versus Discrete-time Approaches for Scheduling of Chemical Processes: a Review.
Computers & Chemical Engineering, 28, 2109-2129, 2004.
Giannelos, N. F.; Giorgiadis, M. C. A Simple New Continuous-Time Formulation for Short-Term Scheduling of Multi-
purpose Batch Processes. Industrial & Engineering Chemistry Research, 41, 2178-2184, 2002.
Fonseca et al (2018)
22
Hakama, A.; Khouloud, M.; Zeroual, Y. Manufacturing of superphosphates SSP & TSP from down stream phosphates.
Procedia Engineering, 46, 154-158, 2012.
Ierapetritou, M. G.; Floudas, C. A. Effective Continuous-Time Formulation for Short-Term Scheduling. 1. Multipurpose
Batch Processes. Industrial & Engineering Chemistry Research, 37, 4341-4359, 1998.
Ierapetritou, M. G.; Floudas, C. A. Effective Continuous-Time Formulation for Short-Term Scheduling. 1. Continuous
and Semicontinuous Processes. Industrial & Engineering Chemistry Research, 37, 4360-4374, 1998b.
Ierapetritou, M. G; Hené, T. S.; Floudas, C. A. Effective Continuous-Time Formulation for Short-Term Scheduling. 3.
Multiple Intermediate Due Dates. Industrial & Engineering Chemistry Research, 38, 3446-3461, 1999.
IPNI, International Plant Nutrition Institute. Phosphorus fertilizer production and technology by Dr. Mike Stewart.
http://www.ipni.net/article/IPNI-3187, accessed in 11/24/2015.
Janak, S. L.; LIN, X.; Floudas, C. A. Enhanced continuous-yime unit-specific event-based formulation for short-term
scheduling of multipurpose batch processes: resource constraints and mixed storage policies. Industrial & Engineering
Chemistry Research, 43, 2516-2533, 2004.
Karimi, I. A.; McDonald, C. M. Planning and scheduling of parallel semi-continuous processes. Part 2. Short-term scheduling.
Industrial & Engineering Chemistry Research, 36, 2701– 2714, 1997.
Kondili, E.; Pantelides, C. C.; Sargent, R. W. H. A general algorithm for short term scheduling of batch operations- MILP
formulation. Computers & Chemical Engineering, 17, 211-227, 1993.
Loureiro, F. E. L.; Monte, M. B. M., Nascimento, M. Industrial Rocks and Minerals, CETEM, c.7, 141, 2005.
Mangwandi, C.; Albadarin, A. B.; Al-Muhtaseb, A. H.; Allen, S. J.; Walker, G. M. Optimization of high shear granulation
of multicomponent fertilizer using response surface methodology. Powder Technology, 238, 142-150, 2013.
Maravelias, C. T.; Grossmann, I. E. New general continuous-time State-Task Network formulation for short-term scheduling
of multipurpose batch plants. Industrial & Engineering Chemistry Research, 42, 3056-3074, 2003.
Méndez, C. A.; Cerdá, J.; Grossmann, I. E.; Harjunkoski, I.; Fahl, M. State-of-the-art review of optimization methods for
short-term scheduling of batch processes. Computers & Chemical Engineering, 30, 913-946, 2006.
Pantelides, C. C. Unified frameworks for optimal process planning and scheduling. In D. Rippin, J. Hale, & J. Davis
(Eds.), Proceedings of the second international conference on foundations of computer-aided process operations, 253–274,
1994.
Research and Marketing. Phosphate Fertilizers Market by active ingredient, crop type & by geography: trends forecast
to - Energy and Power: by marketsandmarkets.com. Publishing Date: April 2014, Report Code: AGI 2403.
Shaik, M. A.; Floudas, C. A. Improved Unit-Specific Event-Based Continuous-Time Model for Short-Term Scheduling of
Continuous Processes: Rigorous Treatment of Storage Requirements. Industrial & Engineering Chemistry Research,
46, 1764-1779, 2007.
Shaik, M. A.; Floudas, C. A. Unit-specific event-based continuous-time approach for short-term scheduling of batch
plants using RTN framework. Computers & Chemical Engineering, 32, 260–274, 2008.
Shaik, M. A.; Janak, S.L.; Floudas, C. A. Continuous-Time Models for Short-Term Scheduling of Multipurpose Batch
Plants: A Comparative Study. Industrial & Engineering Chemistry Research, 45, 6190-6209, 2006.
Sundaramoorthy, A.; Karimi, I. A. A simpler better slot-based continuous-time formulation for short-term scheduling in
multipurpose batch plants. Chemical Engineering Science, 60, 2679-2702, 2005.
Velez, S.; Maravelias, C. T. Multiple and nonuniform time grids in discrete-time MIP models for chemical production
scheduling. Computers & Chemical Engineering, 53, 70-85, 2013.
Velez, S.; Maravelias, C. T. Theoretical framework for formulating MIP scheduling models with multiple and non-uniform
discrete-time grids. Computers & Chemical Engineering, 72, 233-254, 2015.
Journal of Applied Operational Research Vol. 10, No. 1
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Appendix: Input data
Capacity and changeover data provided in this appendix is given in an hourly basis. Depending on the adopted period
length data must be adjusted accordingly.
Table A1. Processing Units Capacity (product depend).
Unit Product Capacity (tons/h)
Sulfuric acid plant Sulfuric acid 150
Prosphoric acid Reactor Phosphoric acid 80
Prosphoric acid Evaporator MAP phosphoric acid 80
TSP phosphoric acid 70
Acidulation SSP 120
TSP 120
Granulation – Unit 1
MAP 80
SSP 90
SSPA 140
TSP 130
Granulation – Unit 2
MAP 120
SSP 90
SSPA 70
TSP 90
Granulation – Unit 3
MAP 120
SSP 60
SSPA 70
TSP 80
Table A2 – Data relative to batch processes considering each warehouse.
Warehouse Stall Curing time
(h)
Minimum Load
(tons)
Maximum Load
(tons)
Warehouse 1
1 120 300 700
2 120 300 700
3 120 300 700
4 120 300 700
5 120 300 700
Warehouse 2
1 144 300 900
2 144 300 900
3 144 300 900
4 144 300 900
5 144 300 900
Table A3. Initial inventory by intermediate or end product.
Product Quantity (tons)
Sulfuric acid 4,000
TSP 100
TSP phosphoric acid
MAP phosphoric acid
Cured SSP in Warehouse 1, Stall 1
Cured TSP in Warehouse 2, Stall 1
100
200
700
900
Fonseca et al (2018)
24
Table A4. Changeover time (h).
Unit Product
Acidulation
SSP TSP
SSP 0 8
TSP 8 0
Granulation
(Units 1, 2, 3)
MAP SSP SSPA TSP
MAP 0 16 16 16
SSP 8 0 8 8
SSPA 8 8 0 8
TSP 8 8 8 0
Table A5. Gross Profit.
Product Gross profit (R$/ton)
SSP
SSPA
200
230
TSP
MAP
300
250
Table A6. Consumption factors, which represents tons of input material per ton of output product.
Task Product Consumption factor
H2SO4 Reaction Sulphur 0.33
Water 0.17
H3PO4 Reaction
Fine Concentrate 0.50
Dry Concentrate 2.50
H2SO4 2.50
H3PO4 Evaporation (MAP) H3PO4 1.00
H3PO4 Evaporation (TSP) H3PO4 1.00
Acidulation (SSP) H2SO4 0.40
Fine Concentrate 0.60
Acidulation (TSP) Dry Concentrate 0.45
H3PO4 (TSP) 0.40
Granulation (MAP) NH3 0.14
H3PO4 (MAP) 0.54
Granulation (SSP) Cured SSP 0.75
Granulation (SSPA) Cured SSP 0.75
NH3 0.05
Granulation (TSP) Cured TSP 1.00