an milp production scheduling model for a phosphate ... · the production schedule was represented...

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]

Journal of Applied Operational Research Vol. 10, No. 1

3

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)

4

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


5

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


6

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.


7

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.


8

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


9

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


10

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


11

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


12

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)


13

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


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


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)


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

400

600

800

1 10 19 28 37 46 55 64 73 82



0

200

400

600

800

1 10 19 28 37 46 55 64 73 82



0

200

400

600

800

1 10 19 28 37 46 55 64 73 82



0

200

400

600

800

1 10 19 28 37 46 55 64 73 82



0

200

400

600

800

1 10 19 28 37 46 55 64 73 82



0

200

400

600

800

1 10 19 28 37 46 55 64 73 82



0

200

400

600

800

1 10 19 28 37 46 55 64 73 82



0

200

400

600

800

1 10 19 28 37 46 55 64 73 82



0

200

400

600

800

1 10 19 28 37 46 55 64 73 82




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

400

600

800

1000

1200

1400

1600

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


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


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



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



Loading (Stall)

Curing (Stall)

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



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



Loading (Stall)

Curing (Stall)

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 91

92

93

94

95

96

97

98

99

100

101

102

103

104

105

106

107

108

109

110

111

112

113

114

115

116

117

118

119

120

121

122

123

124

125

126

127

128

129

130

131

132

133

134

135

136

137

138

139

140

141

142

143

144

145

146

147

148

149

150

151

152

153

154

155

156

157

158

159

160

161

162

163

164

165

166

167

168

169

170

171

172

173

174

175

176

177

178

179

180

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)


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


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.


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.


23

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


MAP 120

SSP 90

SSPA 70

TSP 90


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


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

an milp production scheduling model for a phosphate ... · the production schedule was represented...

Documents