Optimal Multi-Site Resource Allocation and Utility Planning for

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Optimal Multi-Site Resource Allocation and Utility Planning for Integrated Rice Mill Complex Jeng Shiun Lim , Zainuddin Abdul Manan,* Haslenda Hashim, and Sharifah Rafidah Wan Alwi Process Systems Engineering Centre (PROSPECT), Faculty of Chemical Engineering, Universiti Teknologi Malaysia, 81310 Skudai, Johor, Malaysia S Supporting Information *

ABSTRACT: A rice mill produces rice and a variety of by-products, including broken rice, rice bran, and rice husk. Rice husk, in particular, has been widely utilized as a source of fuel to generate heat and electricity. Efficient conversion of the by-products into value-added products via an integrated resource-efficient (IRE) rice mill complex provides a strategic option for rice mills to remain competitive. However, in the face of increasing global demand for rice, optimal resource allocation has become a major challenge for rice enterprises, which operate multiple rice mills with different technologies and capacities. This work proposes a Multi-Site Resource Efficient (MSIRE) System, which is a model-based generic framework for the efficient management of the aforementioned complex, interacting issues. The MSIRE system has been proposed to (i) plan resource allocation and utility network at each site, (ii) screen and select the most suitable technology, (iii) determine the logistic network, and (iv) plan capacity expansion with the objective of maximizing overall profit. The eight different scenarios analyzed with the MSIRE framework demonstrate the tool’s powerful capability to guide planners toward a profitable rice mill business. Because of its generic element, the MSIRE can also be applied to other types of industry.

1. INTRODUCTION Rice is an important staple food for almost half of the world’s population.1 It is estimated that between 2000 and 2050, the global rice consumption will increase by 35%.2 To cope with the increasing demand, rice enterprises are expected to expand their current rice processing capacities. The expected increase in production costs due to increase in capacity may require rice enterprises to be resource efficient and to generate additional revenues by converting their rice mills’ by-products into valueadded products and sources of energy. These goals can be achieved by transforming the current rice mills into integrated resource-efficient (IRE) rice mill complexes. An IRE rice mill complex integrates its various processes to efficiently utilize its resources and to produce fuel, electricity, and value-added products. As shown in Figure 1, the supply chain network of a rice enterprise with an IRE rice mill complex consists of multi-stage and multi-site elements. The stages include paddy fields, process sites, and distribution centers. The process sites can consist of substages or process units, which include the drying facility, milling facility, and other downstream processes. Conversely, the site consists of clusters of stages at different geographical locations. The supply chain begins with the transportation of wet paddy from the paddy fields to the process site. The wet paddy is then dried in the drying facility at the process site. The dried paddy is then fed to a rice mill where it is manufactured into full head rice, broken rice, rice bran, and rice husk. The full head rice and broken rice are mixed in a rice packaging plant to produce graded rice of different compositions. Downstream of the rice mill, the by-products are processed into value-added products and utilized as sources of energy. For instance, rice husk is utilized as a biomass fuel in the cogeneration system, an inherently cleaner production compared to fossil fuel-based energy sources.3 Such utilization © 2013 American Chemical Society

prevents the negative environmental and social impacts associated with the conventional way of disposing rice husk.4 The by-products are then transferred to the distribution centers at strategic locations. Note that an enterprise can operate a few process complexes and distribution centers at multiple sites, which provides the enterprise with the flexibility of producing its products at different locations, as opposed to the standard practice of centralizing production at a single site. To maximize the economic benefits of an IRE rice mill complex, the enterprise may need to reconfigure its current resource allocation strategy and the corresponding utility network at both the stage and site levels. At the stage level, the enterprise must select a profitable production portfolio and the corresponding technology, while satisfying the variability supply and demand constraints of the IRE rice mill complex. Conversely, dealing with rice mills at multiple sites necessitates proper coordination among the various sites, subject to the facilities’ capacity constraint. The specific tradeoffs involved in IRE rice mill complex are as follows: (1) Variability in energy demand: The harvested paddy must be dried within 48 h to prevent degradation. Therefore, the rice mill will predominantly require thermal energy for drying during the harvesting season. (2) Energy supply options (cyclonic husk furnace vs cogeneration system): The cyclonic husk furnace (CHF) can provide a relatively cheaper thermal energy option compared to the cogeneration system. While the CHF can directly generate hot air for dryers, the cogeneration Received: Revised: Accepted: Published: 3816

October 22, 2012 December 25, 2012 February 1, 2013 February 1, 2013 dx.doi.org/10.1021/ie302884t | Ind. Eng. Chem. Res. 2013, 52, 3816−3831

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Figure 1. General supply chain network for rice industries.

All of these issues highlight the need for a systematic framework to plan the optimal resource allocation and the utility network for a multi-site IRE rice mill complex. Multi-site production planning in processing and manufacturing industries has received significant attention from researchers. In the manufacturing industry, Guinet proposed the primal-dual approach and tested this approach with a wide range of problems to minimize variable and fixed costs.5 This model was used for global production planning and local workshop scheduling. Levis and Papageorgiou later presented a systematic mathematical programming framework for longterm multi-site capacity planning with uncertainty for the pharmaceutical industry.6 In the petroleum refining industry, Kim et al. proposed a model that integrates the supply network with production planning for multi-site refineries producing multiple products to examine the effects of reallocating distribution centers. The results show that the reallocation of the distribution center saves costs.7 Pitty et al. considered the stochastic variation in transportation, yields, price, and operational problems in formulating an integrated refinery supply chain.8 Verderame and Floudas developed a model for multi-site planning with production disaggregation that was aimed at executing the operational-level decisions to increase the efficiency of production facility utilization and customer-order fulfilment over a time horizon of several months.9 Recently, Kopanos and Puigjaner formulated a general mixed-integer linear programming (MILP) model based on the key aspects of scheduling/batching and production planning to identify the optimal scheduling decisions for a multi-site batch process system. In this work, the batch sizes and processing times were simultaneously optimized with the scheduling decisions.10 Narodoslawsky et al. highlighted several key challenges in the biomass supply chain planning and management. These include the competition posed by other resources, the need to identify the optimal locations as well as the corresponding capacities of process plants, issues of process synthesis, and related technological challenges.11 To address these challenges, there have been several recent works related to the biomass supply chain. For instance, van Dyken et al. formulated a mixed-integer linear model to represent the key components of the biomass supply chain that includes supply and demand, as well as storage and processing considerations. The model correlates the energy

system involves more losses as it channels the heat from the turbine exhaust steam into the air radiator to convert the energy source into hot air. However, the cogeneration system is able to generate on-site electricity while providing the thermal energy in the form of steam. Apart from satisfying the heat demand for the drying system, steam from the cogeneration system is also required in the rice bran oil extraction process. Therefore, to design an optimal utility system using rice husk as the fuel, the trade-off between energy efficiency and cost-effectiveness of the various energy alternatives needs to be carefully considered. (3) Rice husk limitation: During the drying period, there may not be sufficient rice husk in a rice mill to satisfy the extensive thermal energy requirement of the dryers. Hence, it may be necessary to purchase and transport rice husk from other rice mills. The rice enterprise needs to decide whether to purchase rice husk from another rice mill to satisfy the requirements of the CHF or cogeneration system or to directly buy electricity to satisfy both heat and electricity demand. Thus, there is a trade-off between the costs of transportation, purchase of additional rice husks, and electricity cost. (4) Centralized vs decentralized process complex: Harvested paddy and rice husk may be transferred to a centralized drying facility that has a large-scale cogeneration system. A decentralized drying facility may also be employed as an alternative. In this case, several smaller-scale cogeneration systems may be installed at selected rice mills. Ultimately, selection of either a centralized or a decentralized drying facility essentially involves a trade-off between the capital and the logistic costs. (5) Rice mill capacity expansion vs rice mill construction on a new site: An enterprise can capitalize on its existing infrastructure and save on the capital cost by choosing to expand a rice mill’s capacity at a current site. From the perspective of logistic, however, the existing process complex may not be the most strategic site. The trade-off between the logistic cost to transport the resources and the capital cost needs to be evaluated. 3817

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content in the biomass and its moisture level.12 Bowling et al. formulated a mathematical model to determine the capacity, operational strategies, and locations of the biorefinery and pretreatment facilities.13 The model is capable of determining the optimal supply chain configuration that includes the site selection (centralized or decentralized) and the biomass feedstock location in order to maximize the overall profit. Another noteworthy study is conducted by You and Wang. The authors considered the key factors in supply chain planning and management such as the conversion pathways, feedstock seasonality, geographical diversity, resource degradation, and demand distribution.14 In another study, Č uček et al. formulated a regional biomass supply chain model with the objective function to maximize profitability. The model is based on a four-layer superstructure that includes harvesting, preparation, core processing, and product distribution as the key components.15 Leduc et al. developed a model to identify the optimal locations of lignocellulosic ethanol refineries integrated with polygeneration.16 Later, Zhang et al. formulated an optimization model for a switchgrass-based bioethanol supply chain. The objective of the model is to determine the logistic decision by minimizing the overall cost.17 Zhang et al., on the other hand, developed a model for biofuel supply chain. The model considers the key elements such as biomass harvesting and processing, transportation, and storage to identify the optimal location for biofuel process sites and corresponding logistic networks.18 Marvin et al. proposed a model that considers the existing ethanol production facilities from corn and the new potential sites for biofuel production. The minimum selling price for the biofuel at each facility is determined by performing a detailed cash flow analysis. Recently, Wang et al. presented an energy crop supply chain model to identify the optimal location and capacities for a cogeneration facility, subject to the minimum cost of the overall system.19 For the pulp and paper industry, Aksoy et al. analyzed four different technologies to convert woody biomass and mill waste into bioenergy products from the perspective of feedstock allocation, facility location, and economic performance.20 For the rice industry, Delivan et al. performed a supply chain analysis on a rice−straw-based cogeneration system that involves harvesting, loading, transporting, storage, and process for energy exploitation.21 However, the authors did not consider the integration of cogeneration system with the rice mill. Later, Lim et al. proposed a model for the optimal design of a rice mill utility system that includes the cogeneration system and CHF to satisfy the heat and electricity requirements of the rice mill throughout the year. To achieve the minimum cost, the model also considers the optimal planning of the rice husk logistic network.22 This study, however, does not include the potential of converting the rice mill by-products into value-added products. An analysis of the existing literature shows that only one study has focused on modeling the rice supply chain.23 The contributions of that study were two-fold. (i) It identified the drying and storage capacities as the bottlenecks of rice milling. (ii) It explored various scenarios involving the capacity adjustments of the drying and milling processes and evaluated the economic performances of these scenarios. However, there are a few noteworthy limitations of the study. First, it only considered a single product. Second, the process technology was predefined by the user. Finally, the utility system was not considered. Thus, the opportunity of utilizing the by-products from the rice mill to generate commercial product or energy products has been overlooked. There is a clear need for a systematic approach to optimize the rice mill supply chain by

integrating production planning with utility network, along with technology selection, capacity planning, and site selection. To address these needs, an integrated model for a rice mill’s production and utility planning has been formulated and presented in this paper. The model enables an enterprise to perform simultaneous resource allocation as well as to plan capacity expansion for rice mills that are located at multiple sites with the aim of maximizing overall profit. This paper begins by presenting a superstructure that describes the MILP problem (Section 2) prior to the development of the Multi-Site Resource Efficient (MSIRE) model (Section 3). The model is tested using eight different scenarios (Section 4). Then, the impacts and sensitivities of the various scenarios on a rice mill’s profitability are analyzed (Section 5). 1.1. Problem Statement. A rice mill enterprise operates two rice mills (Rice Mills A and B) in Northern Malaysia. Currently, these rice mills have a combined annual drying capacity of 50,000 tonnes of paddy harvested from several paddy fields in the region. The heat sources of the rice mills’ drying facilities are supplied by CHF, which is fuelled by rice husk. The main product of the rice mills is graded rice, which is a mixture of full head rice and broken rice. The unused broken rice and by-products from the rice milling process, such as rice husk and rice bran, are sold directly to downstream industries. To cope with the market demand and generate more profit, the rice enterprise has outlined two strategies, including (i) transforming the rice mill into an IRE rice mill complex and (ii) expanding the capacity of the rice mill facilities. By transforming the conventional rice mill into an IRE rice mill complex, the byproducts can be utilized as feed to produce value-added products or as a renewable fuel for the mill’s utility system. For instance, the unused broken rice can be ground into rice flour using a flour-grinding machine. Furthermore, rice bran oil can be extracted from the rice bran using the appropriate technology for rice bran oil extraction. Rice husk can be consumed as a renewable fuel for the CHF or the cogeneration system. Along with the CHF, the cogeneration system can fully satisfy the heat and part of the electricity demands of the corresponding rice mill. To fulfill the fuel demands of the cogeneration and CHF systems during peak periods, it may be necessary to purchase and transport additional rice husk from other rice mills. Conversely, the expanded process facilities are expected to meet the increase in demand, which can be achieved by expanding the current process facilities in Rice Mills A and B. Alternatively, a new facility can be constructed at Rice Mills C or D. Note that these locations are selected by the rice enterprise, as they are situated close to the paddy field and the other companies’ rice mills, which can be a source of rice husk. Given the different sets of process and operating data for a technology (feed material composition, product/energy conversion yield, specific energy consumption, available capacities, and seasonal operating hours), the logistic data (distances linking paddy fields, process sites, and distribution centers), and the cost data (product price, material cost, utility cost, transportation cost, and capital cost), the problem consists of evaluating the following: (i) the product portfolio for each rice mill, (ii) the utility system’s location and scale, (iii) the decision of whether to expand the current processing facility or build a new facility, and (iv) the configuration of the paddy and rice husk logistic network. The relevant variables to be determined include the continuous and binary variables. The continuous variables are the quantity of resource/utility intake, the quantity of raw materials fed into a process unit, and the corresponding product quantity. 3818

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Figure 2. Pathway of resource i at process site pc.

(see “resource i” box in Figure 2). In the context of this paper, S.G. resource (SGRES t,pc,p,m,i) refers to the material generated under the operating mode m of process p within the same process site pc during period t. The incoming intermediate resource (IMRES t,pc,i) is the resource transported from another process site pc during period t. The external resource (EXRESt,f,pc,i) is the resource purchased from other companies. Conversely, there are a few pathways that utilize the resources at the process site. First, the resource can be further utilized as feeds for other processes within the same process site (MAT t,pc,p,m,i). The resource can also be transferred to other process sites (pc′) as an outgoing intermediate resource (TRANSRES t,pc,pc′,i). Additionally, the resource can be sold directly as a by-product (BYPRO t,pc,i) from process site pc or transferred to distribution center c as a product (TRANSPRO t,pc,s,i). Unused resources can be shifted to the next period (t + 1) as an inventory (INVRES t,pc,i). The classes of utilities include the utility generated on-site (SGU t,pc,p,m,p,u), the utility converted from other types of utilities (SCU t,pc,p,u), and the external utility (EXU t,pc,u). The utilities can be consumed to satisfy the needs of the processes (UDEM t,pc,p,u) or can be fed into a process (INU t,pc,p,u) and converted into other types of utilities. Note that there may be an excess amount of utilities generated within the system (EXCESSU t,pc,u).

The decision variables are (i) technology, (ii) capacity, and (iii) site location. The decisions are to fulfill product demand while considering the availability of resources from various locations. The next section describes the superstructure representing the formulated MSIRE model.

2. SUPERSTRUCTURE DEVELOPMENT A generalized MSIRE model has been developed with several key functionalities: (a) Resource allocation planning used to either to sell the resource directly or to further process it into a value-added product. (b) Logistic planning used to identify the logistic network interlinking the source location, process site, and distribution center. (c) Design of the IRE complex used for selection of the processes and appropriate capacities for the process and storage facilities. Subsequently, Figure 2 shows the superstructure of the multisite IRE rice mill complex. The process stages in this figure include the external location f (paddy field and external rice mill), process site pc (process complex at the existing and new sites), and the distribution center s, which are all situated at different locations. The line indicates the transhipment of resource i either between the aforementioned locations of interest or between two processes p within the same process site. The red dotted line represents the flow of utility u in a process site during period t. Note that resource i refers to the feed from the process or the subsequent product generated from the corresponding process. Depending on its functionality and source of intake, resource i can be termed as the system-generated (S.G.) resource, incoming intermediate resource, or external resource

3. MODEL FORMULATION The elements of the framework are highlighted in following sections: (i) Generic: The framework and model are generic and can be applied to other industries with minor modifications. 3819

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RCOST represents the total material cost, where EXRESt,f,pc,i is the input rate of the external resource i from location f, while URCOSTi denotes the unit purchasing cost of the corresponding external resource i. RCOST is given by eq 4.

In some cases, a constraint equation may not apply to all situations. To maintain the generic nature of the equation, a binary parameter is associated with specific terms in the equation. The parameter will act as an indicator to allow the corresponding term to be switched on/off. Apart from the aforementioned issue, note that the input composition and output conversion yield for the functional units may be different from one another. Therefore, to handle these two features, various types of matrices and indicators are incorporated into the material balance equation sets. (ii) Model formulation and time representation: The model is formulated in a multi-period where the annual time horizon is divided into 12 months. Note that the purpose of this model is focused on resource allocation, capacity, and logistic planning for a long time horizon as opposed to day-to-day scheduling. (iii) Assumptions: • No storage charges are considered for the inventory, as the function of the proposed model is mainly for strategic long-term resource allocation planning and technology screening rather than short-term scheduling. • A fixed electricity load factor is not considered. • The costs of piping and steam distribution system are considered negligible compared to the equipment cost. • No binary variable has been defined for the selection of the paddy field location or distribution center because the number of sites does not need to be fixed. In addition, there is no fixed cost associated with these locations, as the incurred cost is solely due to the logistic cost, which depends on the type of resources and the distances between these locations and rice mill complexes. 3.1. Objective Function. The objective function of the model is to maximize the overall profit (PROFIT), as described by eq 1. The function consists of the revenue (REV), by-product revenue (BYREV), material cost (RCOST), processing cost (PCOST), total capital cost (CCOST), and logistic cost (LCOST).

RCOST = NOS ×

PCOST = NOS ×

UCOST = NOS ×

MATt , pc , p , m , i × UPCOSTp (5)

∑

EXUt , pc , u × UUCOSTu (6)

t , pc , u

LCOST is the logistic cost, which consists of several components, including TCEXRESt,i (incurred transportation cost between the resource supply location and process site), TCIMRESt,i (incurred transportation cost between two process sites at different sites), and TCPROt,i (incurred transportation cost between the process site and distribution center). LCOST is defined by eq 7. LCOST = NOS × (∑ TCEXRESt , i + TCIMRESt , i i

+ TCPROt , i)

(7)

CCOST refers to the annualized capital cost involved in expanding the current process capacities or installing a new process/technology at a site. CCOST is defined in eq 8, where YEXPp,pc,z is a binary variable used in determining the expansion capacity of process p at process site pc with size z, and EXPCOSTp,z is the corresponding annualized expansion cost. CCOST =

∑

YEXPp , pc , z × EXPCOSTp , z (8)

p , pc , z

The final term, FCOST, is related to the annualized cost to construct (CSTCOSTpc) process complex pc at a new site. YPCpc is a binary variable used to decide whether process complex pc will be constructed at the new site.

REV (revenue) is represented by eq 2, where NOS is the number of seasons in a year, and PIi is the decision parameter to indicate the feasibility of selling the resource i as a product, while PROt,s,i is a variable that denotes the quantity of the distributed product i at distribution center s during period t. PRIi denotes the unit price of product i, which is obtained from the published price.

FCOST =

∑ YPCpc × CSTCOSTpc (9)

pc

∑ PIi × PROt ,s ,i × PRIi

3.2. Mass Balances of Resources. Equation 10 describes the balances of the input sources. RESt,pc,i is the quantity of resource i at process site pc. SGRESt,pc,p,m,i is the systemgenerated resource within the same process site. IMRESt,pc,i is the incoming intermediate resource from other process sites. EXRESt,f,pc,i is the external resource.

(2)

BYREV represents the revenue generated from the selling of by-product i. BYREV can be expressed by eq 3 in terms of BYPRO t,pc,i, which is the by-product i at process site pc. BIi represents the decision parameter to indicate the feasibility of selling resource i as a by-product.

∑ SGRESt , pc , p, m, i + IMRESt , pc , i + ∑ EXRESt , f , pc , i = RESt , pc , i p,m

∑ BIi × BYPROt ,pc ,i × PRIi t , pc , i

∑

UCOST represents the total utility cost as defined by eq 6, where EXUt,pc,u is the external utility requirement of utility u at process site pc during period t, and UUCOSTu is the relevant utility cost.

(1)

BYREV = NOS ×

(4)

t , pc , p , m , i

− UCOST − LCOST − CCOST − FCOST

t ,s,i

EXRESt , f , pc , i × URCOSTi

PCOST represents the total cost involved in producing the resource via the corresponding technology (eq 5). MATt,pc,p,m,i denotes the input rate of material i into technology p under operating mode m at processing complex pc. The UPCOSTp represents the unit processing of process p.

PROFIT = REV + BYREV − RCOST − PCOST

REV = NOS ×

∑ t , f , pc , i

f

∀ i ∀ pc ∀ t

(3) 3820

(10)

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(SGRES t,pc,p,m,i) or a system-generated utility (SGU t,pc,p,m,u). The integration matrices, including the material composition matrix (MCMi, m, p), process resource conversion matrix (PRCM p, m, i ), and process utility conversion matrix (PUCMp, m, u), are three important indexes of the corresponding process, and they represent the material composition, product conversion, and utility generation of a particular process, respectively.

Note that the intake of an external resource is governed by the availability of the corresponding resource. This constraint is formulated in eq 11, where AVAIRESt,f,i is the available quantity of resource i at location f during period t. AVAIRESt , f , i >

∑ EXRESt ,f ,pc ,i

∀f ∀i ∀t (11)

pc

Note that resources can be utilized as materials within the same process site as by-products to be sold directly from the process site, an intermediate resource to be utilized in another process site, or a product to be sent to a distribution center, as described in following set of equations (eqs 12−15). In these equations, INVRESt,pc,i is the quantity of resource i at process site pc during period t that shifts to period t + 1 as inventory, while OPEINVpc,i is the initial inventory at the beginning of period t = 1. Note that INVi is the inventory indicator used to denote the feasibility of shifting resource i into the next period. TRANSRESt,pc,pc′,i is the transhipment flow of intermediate resource i between a process site pc and another process site pc′ during period t. TRANSPROt,pc,s,i is the transhipment flow of product i between process site pc and distribution center s during period t.

∑ TRANSRESt ,pc ,pc′ ,i = IMRESt ,pc ′ ,i

MATt , pc , p , m , i = PRESt , pc , p , m × MCMi , m , p ∀ i ∀ p ∀ m ∀ pc ∀ t

(18)

PRESt , pc , p , m × PRCM p , m , i = SGRESt , pc , p , m , i ∀ i ∀ p ∀ m ∀ pc ∀ t

(19)

PRESt , pc , p , m × PUCM p , m , u = SGUt , pc , p , m , u ∀ u ∀ p ∀ m ∀ pc ∀ t

(20)

The generated utility can be converted into another type of utility. For instance, the steam generated from the cogeneration system can be converted into hot air via an air radiator. The following equations represent the conversion process, where INUt,pc,p,u is the quantity of the utility resource as an input into the conversion process p, PUt,pc,p is the quantity of the processing utility resource, and SCUt,pc,p,u is the quantity of the system converted utility. The input and output of the utility-converted process are correlated with the utility composition matrix (UCMu,p) and utility−utility conversion matrix (UUCMu,p). INUt , pc , p , u = PUt , pc , p × UCMu , p

∀ i ∀ pc′ ∀ t

∀ u ∀ p ∀ pc ∀ t (21)

pc

(14)

∑ TRANSPROt ,pc ,s ,i = PROt ,s ,i

PUt , pc , p × UUCM p , u = SCUt , pc , p , u

(22)

∀i ∀s ∀t (15)

pc

3.4. Capacity Selection. The hourly throughput of a processing unit (HRPRES p,pc,t and HRPU p,pc,t) is governed by the lower and upper operating limits of the capacity, as described in eqs 23 and 24. CAPp,z is the available capacity for each process p, and CCAPINp,pc,z is the parameter index that denotes the existing capacities of process p at process site pc, while YOEXPt,pc,p,z, is a decision variable for operating process p with an expanded size z during period t. The lowest operating capacity is constrained by the minimum operating level MINOPEp, which is expressed in terms of the percentage of the total capacity. For instance, the boiler of a cogeneration unit (p = 6) must be operated above 50% of its total capacity to ensure its operability.

Note that only the intermediate material can be transported between two process sites. Products such as graded rice and rice bran oil cannot be transported between two sites, as shown in eq 16.

∑ TRANSRESt , pc , pc′ , i = (1 − PIi) × ∑ TRANSRESt , pc , pc′ , i pc

pc

∀ i ∀ pc ∀ pc′ ∀ t

(16)

One factor that governs the quantity of product i is PRODEMt,s,i, which is the demand of product i at distribution center s during period t and is expressed by eq 17. Product i at distribution center s must fulfill the production demand and must not exceed the limit defined by the product demand indicator, PDIi.

MINOPEp × (∑ CAPp , z × CCAPINpc , p , z Z

PRODEMt , s , i < PROt , s , i < PDIi × PRODEMt , s , i ∀i ∀s ∀t

∀ u ∀ p ∀ pc ∀ t

+

∑ CAPp,z × YOEXPt ,pc ,p,z) < HRPRESt ,pc ,p z

(17)