An MILP production scheduling model for a phosphate fertilizer plant using the discrete time representation

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1 Journal of Applied Operational Research (218) Vol. 1, No. 1, 2 24 ISSN (Print), ISSN (Online) 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 1 February 217 Accepted 9 June 217 Keywords: Mathematical modeling Optimization Phosphate fertilizers Production Scheduling Abstract Fertilizer industries have a strategic importance for the intensification of agriculture and replenishment 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 3 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 5 January 218 Copyright ORLab Analytics Inc. All rights reserved. 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 worldwide food demand requires fertilizer usage. Dawson and Hilton (211) estimate that in 25, only half the population will be fed if global fertilizer production does not rise until then. According to Loureiro et al. (25), 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. (213) addressed cyclone operation optimization for the production of concentrated phosphate rock, while Abdul-Wahab et al. (214) 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. (23). 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 1K Uberlândia/MG, Brazil srgneiro@ufu.br

2 Journal of Applied Operational Research Vol. 1, 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. (26). Besides the modeling aspect, the solution performance of the MILP model proposed by Kondili et al. (23) is too dependent on modeling as well as on the computational resources and solution algorithm used. Mendez et al. (26) 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 23. The total reduction of computational time between the early and later years was several orders of magnitude, from 98 s while exploring 1466 nodes to only.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 representation (Resource Task Network) extends the proposal of Kondili et al. (23) 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 9s 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 (25). 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.

3 4 Fonseca et al (218) 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. (21), Maravelias and Grossmann (23) and Sundaramoorthy and Karimi (25). 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. (24) 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 (23) 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 (24) and Maravelias and Grossmann (23) 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. (24) 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 (21). 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 (25) 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 (22) 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. (26) 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 (21), Castro et al (24), Giannelos and Giordiadis (22), Maravelias and Grossmann (23), Sundaromoorthy and Karimi (25) 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 (22) was not able to determine the optimal solution whereas the approach of Castro et al (21), Castro et al (24), Maravelias and Grossmann (23) and Sundaromoorthy and Karimi (25) 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 (27) proposed an improved approach for the short-term scheduling of continuous processes considering rigorous treatment of storage requirements. The same authors (Shaik and Floudas, 28) also proposed a RTN version of the improved STN version of Ierapetritou and Floudas presented in Shaik et al. (26). Pros and Cons of the discrete time representation According to Floudas and Lin (24), 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

4 Journal of Applied Operational Research Vol. 1, No. 1 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 (213, 215) 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 6% of fertilizer production, with its demand growing 2.4% globally and 4.% in Latin America. China, Russia, India, and the United States represent more than 5% of world consumption, thus pricing such products worldwide. The estimated global consumption is approximately 1, ktons/year (Research and Marketing, 214). 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 P 2 O 5 content in virgin rock (a measure of phosphorus amount in rock) usually varies between 2% and 22% (Kulaif, 29). The P 2 O 5 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, 215). The most marketed raw materials for phosphate fertilizer production are rock phosphate concentrate with 33% 38% P 2 O 5, 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., 199). Phosphate Rock +H 2 SO 4 +H 2 SO 4 SSP Phosphoric Acid CaSO 4 + NH 3 + phosphate rock + NH 3 SSPA TSP MAP Figure 1. Phosphate fertilizers: main products (adapted from INPI, 215). Reaction (1) between phosphate rock with concentrated sulfuric acid and water in the stoichiometric reaction produces agriculturally used phosphogypsum (CaSO 4.nH 2 O), hydrofluoric acid, and phosphoric acid. The latter is also used in the production of phosphate fertilizers. Ca 1 F 2 (PO 4 ) 6 + 1H 2 SO 4 + 1H 2 O 1CaSO 4.nH 2 O + 6H 3 PO 4 + 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:

5 6 Fonseca et al (218) Figure 2. Fertilizer phosphate production flowchart. Simple super phosphate (SSP) - CaH(PO 4 ).2H 2 O: 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. Ca 1 (PO 4 ) 6 F 2 + 7H 2 SO H 2 O 3CaH(PO 4 ).2H 2 O + 7CaSO 4.½H 2 O + 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) - CaH 4 (PO 4 ).H 2 O: 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.

6 Journal of Applied Operational Research Vol. 1, No. 1 7 Ca 1 (PO 4 ) 6 F H 3 PO 4 + 1H 2 O 1Ca(H 2 PO 4 ) 2.H 2 O + 2HF (3) Monoammonium phosphate (MAP) - NH 4 H 2 PO 4 : It results from a reaction between phosphoric acid and ammonia. It is the most frequently sold product. NH 3 + H 3 PO 4 NH 4 H 2 PO 4 (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. (212) have shown that the longer the curing, the better is the P 2 O 5 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.

7 8 Fonseca et al (218) 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

8 Journal of Applied Operational Research Vol. 1, No. 1 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. (26), 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 j s t Tasks Production units States (raw materials, intermediate products, or end products) Time frames

9 1 Fonseca et al (218) Sets I j J b J c J CO J i S cc S bc S cb S p S rm Task i consuming state s Task i producing state s States s consumed by task i States s produced by task i Tasks i that can be performed in unit j j units running batch jobs j units performing continuous tasks j units with changeover j units able to perform task i States s produced by continuous processes and consumed by continuous processes States s produced by batch processes and consumed by continuous processes States s produced by continuous processes and consumed by batch processes States s corresponding to end products 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 Dem s State s, Demand K Changeover Penalty M s 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 A jst 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 D st Product s quantity used to meet demand in period of time t F jst State s quantity accumulated in stall j in time t (before curing) L st State s quantity kept in inventory in time t Purchase quantity of state s in time t P st Binary variables W i,j,t Indicates whether unit j begins processing task i at time t X jt Indicates the presence of uncured material accumulated in stall j in time t Y jt Indicates the presence of cured material stored in stall j in time t Indicates whether no tasks are assigned to unit j in time t with task i being the last task performed Z i,j,t

10 Journal of Applied Operational Research Vol. 1, No 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) 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. (6) (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. (23) 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) Constraints (9) and (1) 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 (8c) (9) (1)

11 12 Fonseca et al (218) 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 ( ). 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). (11) (12) (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. 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 (1). 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,. 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. Constraints (18) track the total consumption of each raw material type s for individual time frames. (14) (15) (16) (17) (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)

12 Journal of Applied Operational Research Vol. 1, No Constraints (2) establishes the mass balance of intermediate products produced by continuous processes and consumed by batch processes. The STN representation in Figure 4 shows that (2) 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 (2) 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. (2) 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 (2), 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) 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)). (23) (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. (23) (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

13 14 Fonseca et al (218) 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 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 % 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 ($) 25,896,. 25,896,. 25,896,. MILP ($) 25,896,. 25,896,. 25,896,. Equations 15,636 15,64 18,2 Continuous variables 5,852 5,852 9,452 Binary variables 4,32 4,32 5,76 Number of iterations 135,69 119,557 99,217 Nodes CPU (s) Relative gap (%)... *Processor: Intel Core i7-286qm 2.5GHz (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

14 Journal of Applied Operational Research Vol. 1, No 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, 8, 8, 8, SSPA 1, 1,2 1,2 1,2 TSP 15, 19, 19, 19, MAP 2, 65, 65, 65, H2SO4 Production2SO2SO2SO2SO2SO2SO2SO2SO2SO2SO2SO2SO2SO2SO2SO2SO2SO2SO2SO2SO2SO2SO2SO2SO2SO2SO2SO2SO2SO2SO2SO2SO2SO2SO2SO2SO2SO2SO2SO2SO2SO2SO2SO2SO2SO2SO2SO2SO2SO2SO2SO2SO2SO2SO2SO2SO2SO2SO2SO2SO2SO2SO2SO2SO2SO2SO2SO2SO2SO2SO2SO2SO2SO2SO2SO2SO2SO2SO2SO2SO2SO2SO2SO2SO2SO2SO2SO2SO2SO2SO H3PO4 ProductionPO4 PO4 PO4 PO4 PO4 PO4 PO4 PO4 PO4 PO4 PO4 PO4 PO4 PO4 PO4 PO4 PO4 PO4 PO4 PO4 PO4 PO4 PO4 PO4 PO4 PO4 PO4 PO4 PO4 PO4 PO4 PO4 PO4 PO4 PO4 PO4 PO4 PO4 PO4 PO4 PO4 PO4 PO4 PO4 PO4 PO4 PO4 PO4 PO4 PO4 PO4 PO4 PO4 PO4 PO4 PO4 PO4 PO4 PO4 PO4 PO4 PO4 PO4 PO4 PO4 PO4 PO4 PO4 PO4 PO4 PO4 PO4 PO4 PO4 PO4 PO4 PO4 PO4 PO4 PO4 PO4 PO4 PO4 PO4 PO4 PO4 PO4 PO4 PO4 PO4 H3PO4 ConcentrationO4-O4-O4-O4-O4O4O4-O4O4O4-O4-O4-O4-O4O4-O4O4-O4-O4O4O4O4O4-O4-O4-O4-O4-O4-O4-O4-O4-O4-O4-O4-O4-O4-O4-O4-O4-O4-O4-O4-O4O4-O4O4-O4-O4-O4-O4-O4-O4-O4O4O4-O4-O4O4-O4-O4-O4-O4-O4-O4O4-O4-O4-O4-O4-O4-O4-O4-O4-O4-O4-O4-O4-O4-O4-O4-O4-O4-O4-O4-O4-O4-O4-O4-O4-O4- AcidulationSSPSSPSSPSSPTSPTSPTSPTSPTSP SSP SSPSSPSSPTSP TSPTSPTSPTSP SSPSSPSSPSSPSSP TSPTSPTSPTSPTSP SSPSSPSSPSSP SSP TSPTSPTSPTSPTSP SSPSSPSSPSSPSSP Stall 1 (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 y w w w w w w w w w w w w w w w 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 Stall 2 (SSP) 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 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 x w w w w w w w w w w w w w w w 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 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 Stall 4 (SSP) 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 y y y 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 Stall 5 (SSP) w w w w w w w w w w w w 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 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 Stall 1 (TSP) w w w w w w w w w w w w w w w w w w w w w w w w w w w w w w 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 x w w w w w w w w w w w w w w w w w w y Stall 2 (TSP) 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 w w w w w w w w w w w w w w w w w w Stall 3 (TSP) 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 w w w w w w w w w w w w w w w w w w y Stall 4 (TSP) 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 Stall 5 (TSP) 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 Granulation 1SSPAMAP MAP MAP MAP MAP MAP SSP SSPA MAP MAP MAP MAP TSPMAP MAP SSPAMAP MAP MAP MAP MAP MAPMAP TSPMAP SSPMAP MAP MAP MAP MAPMAP MAP TSPMAP MAP MAP MAP SSPA MAP MAP MAP MAP MAP TSPTSPTSPSSPA SSPA SSPA Granulation 2TSP MAP MAPMAPMAP SSPA SSPSSP TSP TSPTSP SSPMAP SSPA TSP TSPTSPTSPMAP SSPASSP SSPTSPTSPTSPTSPTSPTSPTSP SSPSSP SSPMAP MAP MAP MAP MAP MAP MAPMAP MAPMAP MAP MAP Granulation 3TSP MAP MAP MAP SSP MAP MAP TSPTSPTSPTSP MAP MAP SSPA SSPA SSPA TSP TSPTSPTSPMAP MAP SSPSSPSSP MAP MAP MAP MAP MAP MAP MAP MAP MAP MAP SSPMAP MAP MAP MAP MAP TSPTSPTSPMAP MAP TSP (Granulation) SSP (Granulation) Loading (Stall) Storage (Stall) H3PO4-MAP (Concentration) SSP (Acidulation) MAP (Granulation) SSPA (Granulation) Curing (Stall) (Concentration) TSP (Acidulation) 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.