Computers and Chemical Engineering 35 (2011) 149–164
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Computers and Chemical Engineering
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Mathematical modeling for simultaneous design of plants and supply chain in the batch process industry
Gabriela Corsano a,b , Jorge M. Montagna a,b,?
a b
INGAR – Instituto de Desarrollo y Dise?o, Conicet – UTN, Avellaneda 3657, (S3002GJC) Santa Fe, Argentina n CIDISI–UTN–FRSF, Argentina
a r t i c l e
i n f o
a b s t r a c t
Most supply chain design models have focused on the integration problem, where links among nodes must be settled in order to allow an ef?cient operation of the whole system. At this level, all the problem elements are modeled like black boxes, and the optimal solution determines the nodes allocation and their capacity, and links among nodes. In this work, a new approach is proposed where decisions about plant design are simultaneously made with operational and planning decisions on the supply chain. Thus, tradeoffs between the plant structure and the network design are assessed. The model considers unit duplications and the allocation of storage tanks for plant design. Using different sets of discrete sizes for batch units and tanks, a mixed integer linear programming model (MILP) is attained. The proposed formulation is compared with other non-integrated approaches in order to illustrate the advantages of the presented simultaneous approach. ? 2010 Elsevier Ltd. All rights reserved.
Article history: Received 5 February 2009 Received in revised form 13 May 2010 Accepted 17 June 2010 Available online 25 June 2010 Keywords: Supply chain design Batch process design MILP Simultaneous optimization
1. Introduction Nowadays, new problems have arisen in business management around enterprises integration, resulting in more complex ?rms with new strategic challenges. They are the consequence of an extremely competitive environment. Two perspectives can be distinguished: ?rst, at the enterprise level, and, second, at the network level. In the ?rst point of view, an enterprise includes different business units that share common resources. Management must build close links among them all with a global approach. Partial solutions optimizing an internal function are not feasible now, and the relationships among subsystems (purchasing, manufacturing, sales, etc.) must be emphasized. In the second perspective, different units or organizations (suppliers, industrial facilities, distribution centers, customers, etc.) collaborate, work together in order to satisfy global objectives. Generally, they do no belong to the same enterprise; they operate in different markets and common solutions affect their operations. Previous efforts focused on ?rm performances, neglecting the overall integration of the network. In this new context, a supply chain (SC) is a usual option where a network of facilities (e.g. plants, warehouses, customers) performs a set of operations ranging from the
? Corresponding author at: INGAR – Instituto de Desarrollo y Diseno, Conicet – ? UTN, Avellaneda 3657, (S3002GJC) Santa Fe, Argentina. E-mail address: mmontagna@santafe-conicet.gov.ar (J.M. Montagna). 0098-1354/$ – see front matter ? 2010 Elsevier Ltd. All rights reserved. doi:10.1016/j.compchemeng.2010.06.008
acquisition of raw materials to the distribution of ?nished goods to customers. Both perspectives must be also integrated in an appropriate decision making structure (Shapiro, 2001). These approaches are valid in all industries, and, particularly, in the Chemical Process Industry where spatial and temporal integration must be addressed (Grossmann, 2005). Varma, Reklaitis, Blau, and Penky (2007) analyze the emerging research challenges around the enterprise-wide modeling and optimization, taking into account that previous Process Systems Engineering efforts have focused on partial and limited subsets of decisions. Several authors have referred to the integration of SC decisions as an important and still open issue on which little work has been published. Reklaitis and McDonald (2004) discussed the importance of integrated SC Management. Also Grossmann (2004) emphasized that the major pending research problem is the integration of planning, scheduling and control, whether at plant level, or at supply chain level. Different problems can be analyzed on the SC: design, planning, scheduling, etc., with different objectives and time horizons. Most of them have been solved through partial approaches where a main decision has been prioritized. However, several authors highlight the advantages of decision making when different perspectives and elements are taken into account in holistic approaches (Bon?ll, ? ? Espuna, & Puigjaner, 2008; Guillén, Badell, Espuna, & Puigjaner, 2006; Puigjaner & Laínez, 2008). For example, Puigjaner and Guillén-Gosálbez (2008) include environmental and ?nancial considerations at the modeling stage in the SC analysis. Amaro and Barbosa-Póvoa (2008) presented a Mixed Integer Linear Program-
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ming (MILP) model for the detailed optimal scheduling of SC where operational decisions are explicitly integrated. Sousa, Shah, and Papageorgiou (2008) presented a two-level planning approach. In the ?rst stage, the global supply chain is redesigned and the production and distribution plan is optimized considering a long time horizon. In a second stage, detailed production and distribution plans are obtained according to the SC con?guration attained from the ?rst stage. Many times, from the resolution point of view, integrated models present a strong challenge: different decisions are dif?cult to be jointly addressed, very large scale optimization problems are posed, and resolution is not easy. Usually, decomposition strategies must be formulated. Focusing on the strategic decision of the SC design, this problem involves making several decisions like number, location, size and installation of plants, warehouses and distribution centres, products to be produced in each plant, allocation of suppliers to plants and plants to distribution centres, etc. There is an abundant and growing body of literature about SC design optimization, including many reviews (Beamon, 1998; Bilgen & Ozkarahan, 2004; Goetschalckx, Vidal, & Dogan, 2002; Klose & Drexl, 2005; Min & Zhou, 2002; Shah, 2005; Shapiro, 2004). Although a detailed review of the literature is beyond the scope of this paper, selected recent works will be mentioned. A lot of attempts have been made to model and optimize the SC design, based on deterministic and stochastic approaches (Amiri, 2006; Applequist, Pekny, & Reklaitis, 2000; Ferrio & Wassick, 2008; ? Guillén, Mele, Espuna, & Puigjaner, 2006; Laínez, Guillén-Gozálbez, ? Badell, Espuna, & Puigjaner, 2007; Santoso, Ahmed, Goetschalckx, & Shapiro, 2005; Thanh, Bostel, & Péton, 2008; Tsiakis, Shah, & Pantelides, 2001; You & Grossmann, 2008). Following the previously cited approach, almost all of them address the SC design problem without analyzing related problems, particularly production plant design. Even though batch process design problem has been an active area of research in the Process System Engineering community over the last decades (Barbosa-Póvoa, 2007), the incorporation of batch plant design in the SC problem has not received much attention. Decision making about plant structures is delayed until SC is con?gured. Shah (2005) points out that both the network and the individual units must be designed appropriately. He also emphasizes that there are potential tradeoffs to be exploited. One of them is manufacturing complexity and ef?ciency related to the number of different products being produced at any site. Despite the extensive background about SC optimization, however, there are no published works dealing with the connection between process design and SC operation (Shah, 2005). Most published works about SC design are posed as distribution problems, where nodes have to be allocated and links among them have to be determined in order to allow an ef?cient operation of the whole system. The logistic aspects have been emphasized rather than the production and operation decisions for the design of these networks (Tsiakis and Papageorgiou, 2008). Two alternatives are predominantly addressed with respect to production plants: they are given (sometimes expansions are considered) or they are black boxes, with a maximum capacity. In the last case, a hierarchical approach is usually employed, where facilities design is postponed. Thus, the tradeoffs between logistic aspects respect to plant performance are not taken into account. A usual problem is the SC design using existing manufacturing facilities (Chen, Wang, & Lee, 2003). For example, Tsiakis et al. (2001) consider the design of a multiproduct, multi-echelon SC with manufacturing sites at ?xed existing locations and where warehouses and distribution centres must be allocated. Production decisions consider the products to be produced at each plant and the plant performance is only taken into account through the
limits on the use of given shared resources. A later work (Tsiakis and Papageorgiou, 2008) integrates new elements using similar assumptions: import duties, exchange rates, plant maintenance, outsourcing production, etc. Taking into account that production facilities are given, there is always a maximum production capacity for any product in each location. Papageorgiou, Rotstein and Shah (2001) present an interesting model for strategic of the SC optimization in the pharmaceutical industry. Even though a holistic approach is employed taking into account product and capacity management, only the allocation of the existing capacity and its expansion are considered. Therefore, the integration of development management with capacity and production planning is a great challenge (Shah, 2004). Ryu and Pistikopoulos (2005) address the design of SC considering three operating policies: coordination, cooperation and competition. Even though the performance of the global system is the main goal, the impact of production facilities is simpli?ed: the capacity of the plants is assumed to be large enough to cover the product demands and is expressed as a linear combination of productions of individual products. Later, Ryu and Pistikopoulos (2007) pose a hierarchical decomposition procedure for a multiperiod planning problem with design considerations. In the ?rst stage, an aggregated model is solved and production requirements for each production facility are determined. In the second step, models are posed for each existing plant to ful?ll assigned demands, and expansion options are available. In this work, a more detailed modeling is presented, where the plants structure must be determined. Unlike past works, the proposed approach incorporates the production plant design together with the SC design and operation in order to attain a more integrated perspective of the SC design problem. Then, plant con?gurations vary, resulting in very different performances related to overall targets of the SC. Therefore, completely different production rates can be attained with a signi?cant effect on the whole SC system. For example, in order to take advantage of production scale, greater facilities are preferred. In contrast, logistic costs are increased. Conversely, when more plants are used, logistic costs are reduced but plants installation costs are increased. Considering multiproduct production facilities, where different recipes are produced in common resources, there are many alternatives and tradeoffs that must be appropriately assessed. Therefore, the simultaneous assessment of all the involved costs allows the adjustment of the plant design to achieve the SC objectives. As previously mentioned, usual procedures include very simple models like black boxes for the plant design and, in a second step, facilities are designed to satisfy speci?c objectives de?ned in the ?rst stage. Thus, the tradeoffs between both problems are missed. The proposed approach solves this question. Experience in previous works shows that bene?ts are attained when holistic approaches are adopted rather than when dealing with decisions separately. In this way, the advantages of integration will be effectively attained. In order to achieve these results, a MILP model is proposed for the simultaneous optimization of SC and multiproduct batch plant designs. Decisions about unit duplication in phase and out of phase and the allocation of intermediate storage tanks are taken into account in order to achieve an ef?cient plant design. Discrete sizes are available for the units, following the usual procurement policy. For comparison purposes, a sequential or hierarchical procedure representing traditional practices is also presented. Within this strategy, SC design decisions are ?rst taken, and plant design is ?tted afterwards considering the SC design previously computed as input parameters. The comparison between the results of the sequential approach and those of the integrated model highlight the advantages of the latter option.
G. Corsano, J.M. Montagna / Computers and Chemical Engineering 35 (2011) 149–164 Table 1 Motivating example: product demands (kg). Customer zones Product I1 I2 I3 K1 150,000 130,000 150,000 K2 100,000 120,000 150,000 K3 115,000 130,000 150,000 Stages J1 J2 Table 2 Motivating example: available unit and tank discrete sizes (l). Available tank size P1 150 150 P2 300 300 P3 500 500 P4 1000 1000 P5 1200 1200 Available tank sizes G1 0 0 G2 500 500 G3 1000 1000 G4 3000 3000 G5
151
5000 5000
In this ?rst approach all the model parameters are deterministic. In practice, however, this is rarely the case as it is usually dif?cult to forecast prices of products and resources, market demands, availabilities of raw materials, etc., in a precise fashion. The uncertainty consideration gives raise to a more complex formulation where the problem is generally tackled through different stochastic scenarios, increasing the number of decision variables and consequently the computational resolution effort. For these reasons, the initial formulation presented in this work is assumed to be deterministic and a stochastic model will be proposed in a future work. The rest of the paper is organized as follows. Section 2 presents a motivating example to highlight the impact of the plant design consideration on the whole SC design and operation. The problem is formulated in Section 3, while the mathematical model is detailed in Section 4. Two studied cases showing the model performance and some comments are posed in Section 5. Finally, the conclusions and future research works are drawn in Section 6. 2. Motivating example Consider a SC comprising raw material sites, production plants and customer zones. There are two potential raw material sites (S1 and S2) with two types of raw materials, namely R1 and R2. Up to three batch plants (L1–L3) can be used to produce three products (I1–I3) and each plant has two batch stages (J1 and J2). For each stage, duplication in phase and out of phase is considered as well as the allocation of intermediate storage tanks between both stages. Unit sizes are selected from a set of ?ve discrete sizes (P1–P5) for batch stages and for storage tanks (G1–G5). Three customer zones (K1–K3) are considered with the product demands shown in Table 1. Table 2 presents the available discrete sizes for units and tanks; Table 3 provides unit size factors and processing times; Table 4 shows distribution cost from raw material sites to plants; and Table 5 indicates distribution cost from plants to
Table 3 Motivating example: batch plant parameters. Size factors J1 I1 I2 I3 0.9 0.6 0.7 J2 0.6 0.5 0.5 Processing times (h) J1 14 12 16 J2 7 8 8
Table 4 Motivating example: distribution cost from raw material sites to plants ($ kg?1 ). Plants Raw material site S1 S2 L1 0.1 0.5 L2 0.3 0.1 L3 0.3 0.3
Table 5 Motivating example: distribution cost from plants to customer zones ($ kg?1 ). Customer zones Plants L1 L2 L3 K1 0.1 0.2 0.3 K2 0.2 0.3 0.1 K3 0.3 0.1 0.2
customer zones. Two different models are proposed in order to evaluate the impact of the plant design on the SC design. The simultaneous approach considers plant design optimization jointly with the SC design. The objective function is the minimization of the total cost given by annualized investment and installation costs of the batch plants, raw material cost, operating cost and distribution cost. The investment cost depends on the unit sizes of each stage in each plant while the installation cost is a ?xed cost independent of production amounts. The raw material cost is
Fig. 1. Motivating example: SC and plant design for the simultaneous optimization approach (amounts × 103 kg).
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Fig. 2. Motivating example: SC design for the sequential approach (amounts × 103 kg).
given by the raw material procurement and depends on the amount of raw material supplied for the different plants. The operating cost is a cost proportional to the production amounts of each product in each plant, and distribution cost considers the amount of product transferred among the different nodes of the SC. The optimal solution consists of only one production plant, L1, with two units out of phase in the ?rst stage and one unit for the second stage, as shown in Fig. 1. The optimal SC design and production amounts (Qil , for product i in plant l) are also shown in Fig. 1. The total cost is equal to $881,000. The sequential or hierarchical approach solves the SC design in a ?rst step. The output decisions from the ?rst step (SC nodes and ?ows between them) are input parameters for the second step where plant design is solved to ful?ll the attained production requirements. The objective function of the ?rst stage minimizes the SC design cost, i.e. the cost associated to the logistic problem: supplied raw material cost, plants installation cost, and distribution cost between SC nodes. The objective function of the second step incorporates the investment cost related to those plants that are allocated according to the solution of the ?rst stage. The solution of this sequential approach for the motivating example consists of two production plants, L1 and L2, and the SC design shown in Fig. 2. Then, giving the SC design as input parameter, the plant design is solved at a second stage. L1 has two units duplicated out of phase at the ?rst stage and one unit in the second stage, while L2 has two units duplicated in phase at the ?rst stage and one unit in the second stage, as shown in Fig. 3. The total cost amounts to $1,007,700, which is 14.38% higher than that of the simultaneous optimization approach solution. Table 6 presents a detailed list of costs for both approaches.
The ?rst stage of the sequential approach minimizes logistic cost. Therefore, the optimal solution uses two production plants for which distribution cost from raw material sites to plants and from plants to customer zones are reduced by 3.1% and 33.5%, respectively, with respect to the simultaneous approach. Plants allocation is prioritized in order to assign facilities near raw material sites and customer zones. Then, in the second step, in order to meet product demands in both plants according to the optimal SC design, more units are required and investment cost is 50% higher than for the simultaneous approach. Due to the in?uence of the investment cost in the objective function, the total cost of the sequential approach is increased. Clearly, the ?rst proposal has considered the tradeoffs between all costs and the scale advantage has prioritized a larger facility against the allocation of several plants near SC nodes. Considering the above results, simultaneous optimization can affect not only the total investment cost but also plant con?guration and SC design as shown in Figs. 1 and 2. This proves that the plant design is operationally and economically critical in the optimal design of the SC. Taking into account the very different values that have been obtained in both solutions, it can be noted that it is no a good option to consider only a ?xed cost for plant installation as usual in the literature. Con?guration and sizing of production plants must be re?ected in the overall objective function. In the following sections, a new formulation is posed for solving the SC design simultaneously with plant performance. 3. Problem statement Four echelons in the SC are considered in this formulation: raw material sites, manufacturing plants, warehouses, and customer zones. At each raw material site s (s = 1,. . ., Ns ), one or more types of raw materials r (r = 1,. . ., Nr ) are available to be delivered to plants l (l = 1,. . ., Nl ), which operate during time horizon Hl . Each multiprod-
Table 6 Detailed list of costs of motivating example ($ year?1 ). Costs Investment Production Raw material Distribution from raw material sites to plants Distribution from plants to customer zones Total Simultaneous approach 349,700 92,200 33,900 169,700 Sequential approach 534,700 116,700 35,400 164,400
235,500
156,500
Fig. 3. Motivating (amounts × 103 kg).
example:
plant
design
for
the
sequential
approach
881,000
1,007,700
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153
Fig. 4. SC representation for the proposed model.
uct batch plant has Njl stages (j = 1,. . ., Njl ) for producing products (i = 1,. . ., Np ). In phase and out of phase unit duplication is allowed as well as the allocation of intermediate storage tanks between two batch stages. They can be allocated in Njl ? 1 positions in plant l, where position j is de?ned between batch stages j and j + 1. A zero wait transfer policy is adopted between consecutive batch stages. A set of Pjl discrete unit sizes SVjl = {VFjl1 , VFjl2 , ..., VFjlPjl } is available for stage j in plant l. In the same way, a set of G jl discrete sizes for storage tanks STFjl = {VTFjl1 , VTFjl2 , ..., VTFjlGjl } is available for position j in plant l. These alternatives follow the usual unit procurement policy. In this way, a linear formulation can be attained. Each available warehouse m (m = 1,. . . , Nm ) has a different stock capacity. Customer zone k (k = 1,. . ., Nk ) has known product demands Dik that must be met. Assuming that the cost parameters associated to plants and warehouses installation, investment, production, distribution, raw material and operation are known, the problem consists of simultaneously settling down: (a) (b) (c) (d) allocation of each plant and each warehouse supplies from each raw material sites production of each product in each plant structure of the plants considering parallel unit duplications and allocation of storage tanks, and unit sizes (e) SC structure: ?ows among SC nodes in order to minimize the total annual cost given by installation, investment, production, operating, and logistic costs. Fig. 4 shows a representation of the SC proposed in this work. 4. Model formulation The problem involves optimizing the SC design simultaneously with the plant design in order to minimize the total cost taking into account and assessing the tradeoffs between plants and supply chain decisions. Following, the basic constraints are posed.
4.1. SC network constraints These constraints are material balances among the different nodes in the SC. Let zil be a binary variable equal to 1 if product i is produced in plant l, and zero otherwise, and Q the material amounts, for example Qil is kg of i produced at l. Following, the constraints between different SC nodes are formulated: 4.1.1. Mass balances between raw material sites and production plants Producing any product at any plant cannot exceed certain limits settled by operative, commercial, and marketing reasons. Therefore, there is always an upper and a lower bound for production capacities:
LO UP zil Qil ≤ Qil ≤ zil Qil
?i, l
(1)
Eq. (1) forces the total amount of product i in plant l to be zero if the product is not produced in l and, otherwise its production is limited. In the same way, each raw material site s has an availability of UP raw material r, Qrs . Then, the use of each resource by the plants is limited by the following expression:
UP Qsril ≤ Qsr i,l
?s, r
(2)
Also, taking into account the product allocation to a plant, each ?ow can be restricted:
UP Qsril ≤ zil Qsr
?s, r, i, l
(3)
Let fril be the raw material conversion factor, i.e. the relationship between the required raw material r and the produced product i. Then, the requirement of resource r in plant l to produce i is
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expressed by
Ns
Qsril = fril Qil
s=1
?r, i, l
(4)
In order to force the production of i in plant l to be zero if the plant does not exist, let exl be the binary variable that takes value 1 if plant l is allocated, and zero otherwise, then zil ≤ exl
?i, l
(5)
4.1.2. Mass balances between production plants and warehouses The total production of i in l has to be delivered to some warehouses m.
Nm
Unlike the general models of batch plant design, in this approach the amounts to be produced in each plant, Qil are process variables. The traditional problem of multiproduct batch plant design is a Mixed Integer Non-Linear Programming (MINLP) model (BarbosaPóvoa, 2007). Here, as was pointed out in the motivating example, the model considers unit and tank sizes selected from a set of standard and discrete sizes that are available to perform the production processes. Using this assumption, a MILP model can be posed (Voudouris & Grossmann, 1992). Also, due to Qil is a model decision, some transformations of variables are needed to keep the linear nature of the problem. Each one of these transformations or variable rede?nitions is appropriately described bellow. The general batch process literature (Biegler et al., 1997) poses the batch unit size of stage j of plant l, Vjl , through a sizing equation which is applied for each product i as follows: Vjl ≥ Sijl Bijl NPjl
Qilm = Qil
m=1
?i, l
(6)
?i, j, l
(11)
Let ym be the binary variable equal to 1 if the warehouse m is max located, zero otherwise, and Qm the capacity of warehouse m. Then
max Qilm ≤ Qm ym i,l
?m
(7)
where Sijl is the size factor (the size required at stage j to produce 1 kg of ?nal product i), Bijl is the batch size of product i at stage j in plant l, and NPjl is the number of in phase units for stage j in plant l. Let Nbijl be the number of batches of product i in stage j of plant l. The amount of product i produced in plant l is de?ned by Qil = Nbijl Bijl
If a warehouse m is allocated, the received products are limited by its capacity. 4.1.3. Mass balances between warehouses and customer zones Assuming that there is not stock accumulation, i.e. steady-state operation, the total amount of product i stored in warehouse m has to be delivered to some customer zones, then
Nl Nk
?i, j, l
(12)
By combining Eqs. (11) and (12) the following constraint are obtained Nbijl ≥ Sijl Qil Vjl NPjl
?i, j, l
(13)
Qilm =
l=1 k=1
Qimk
?i, m
(8)
In order to pose a MILP formulation, non-linear constraints are avoided rewriting the previous design equation in the following manner. Let xxjld be the binary variable that takes value 1 if stage j of plant l has d parallel units in phase, and zero otherwise, so
NP UP
jl
The following constraint forces the amounts delivered from m to be zero if the warehouse m does not exist; otherwise the amount of products to be stored is limited
max Qimk ≤ Qm ym i,k
NPjl =
d=1 NP UP
jl
d xxjld
(14)
?m
(9)
xxjld = exl All the demands have to be ful?lled, so
Nm d=1
?j, l
(15)
Dik =
m=1
Qimk
?i,k
(10)
4.2. Production plant design equations 4.2.1. Batch units The key assumptions considered in this work, which are usual for multiproduct batch plant design (Biegler, Grossmann, & Westerberg, 1997; Voudouris & Grossmann, 1992), are: 1. The size factors and processing times are constant for each product. 2. When multiple parallel units are considered at a stage, they have the same size. 3. The plant operates in single product campaign (SPC) mode throughout the time horizon. 4. When storage tanks are not allocated, ZW (zero wait) policy is employed. 5. If intermediate storage tanks are employed in the process, the operation of the stages upstream is decoupled from those downstream to the tank and the material is not stored for long time, i.e. FIS (?nite intermediate storage) policy is adopted.
UP where NPjl represents the total available units in phase for stage j in plant l. Eq. (15) states that at least one unit per stage must exist if plant l is allocated. The size for unit j of plant l, Vjl , considering the set of available discrete sizes, is given by: Pjl
Vjl =
p=1
vjlp VFjlp ?j, l vjlp = exl ?j, l
(16)
(17)
p ∈ SVjl
Eq. (16) determines the size of unit j in plant l. Eq. (17) forces unit j of plant l to take a discrete size if the plant is allocated or size zero otherwise. Then, substituting Eq. (16) for Eq. (13) and using NPjl de?nition given by Eq. (14) Nbijl ≥
p,d
Sijl Qil VFjlp d
vjlp xxjld
(18)
As constraint (Eq. (18)) is non-linear because of the product Qil
vjlp xxjld , a new nonnegative continuous variable eijlpd is de?ned to
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155
represent this cross-product (Ierapetritou & Pistikopoulos, 1996; Voudouris & Grossmann, 1992): eijlpd = Qil 0 if vjlp and xxjld are equal to 1 otherwise (19)
Nbi,j+1,l ≥ 2
g=1 /
STijl VTFjlg
fijlg
?i, j = 1, 2, ..., Njl ? 1, l
(32)
fijlg ≤ Qil =
UP Qil vtjlg
? i, j = 1, 2, ..., Njl ? 1, l, g ? i, j = 1, 2, . . . , Njl ? 1, l
(33) (34)
Therefore, the following linear constraints are used to represent Eq. (18): Nbijl ≥ Sijl
p,d VFjlp d
fijlg
g
eijlpd
?i, j, l
(20) (21)
UP eijlpd ≤ Qil vjlp d UP eijlpd ≤ Qil xxjld p
?i, j, l, p ?i, j, l, d
If the storage tank does not exist between two consecutive batch stages, then the number of batches must be equal for both of them. Else, the bounds for the ratio between the numbers of batches of consecutive stages can be stated by (Ravemark, 1995) 1 + 1 ?1
g=1 /
vtjlg ≤
Nbi,j+1,l Nbijl
≤ 1 + ( ? 1)
(22) × (23)
g=1 /
Qil =
p,d
eijlpd
?i, j, l
vtjlg ? i, j = 1, 2, . . . , Njl ? 1, l
(35)
4.2.2. Intermediate storage The allocation of an intermediate storage tank between two batch stages causes the process to be decoupled into two subprocesses upstream and downstream of the tank. Therefore, for Njl batch stages there exist, at most, Njl ? 1 possible positions for storage tanks to be allocated between two consecutive batch stages. The capacity constraints for the storage tanks are simpli?ed mass balances around the storage vessels. Different expressions have been proposed. According to Ravemark and Rippin (1998), an upper bound for the storage vessels can be de?ned by VTjl ≥ 2 STijl Bijl sjl
where is a constant value corresponding to the maximum ratio allowed between the number of batches of consecutive stages. It is worth noting that if no storage tank is assigned then vtjlg are equal to zero for all g = 1, so the sums in Eq. (35) are equal to zero and, / therefore, Nbijl = Nbi,j+1,l . Rewriting Eq. (35) as: Nbijl + 1 ?1
g=1 /
vtjlg Nbijl ≤ Nbi,j+1,l ≤ Nbijl + ( ? 1)
(36)
×
g=1 /
vtjlg Nbijl ? i, j = 1, 2, . . . , Njl ? 1, l
? i, j = 1, 2, . . . , Njl ? 1, l ? i, j = 1, 2, . . . , Njl ? 1, l
(24) (25)
VTjl ≥ 2 STijl Bi,j+1,l sjl
where VTjl represents the tank size, STijl the size factor for each storage tank and sjl is a binary variable equal to 1 if a tank is allocated after batch stage j, and zero otherwise. Using Eq. (12) in Eqs. (14) and (25) the storage constraints are: Nbijl ≥ 2 STijl Qil VTjl sjl
and de?ning a new continuous variable ijlg in order to avoid the above non-linearities given by vtjlg Nbijl , the previous equations can be expressed as: Nbijl + 1 ?1
g=1 / ijlg
≤ Nbi,j+1,l ≤ Nbijl + ( ? 1)
?i, j = 1, 2, ..., Njl ? 1, l
sjl
(26) (27)
×
g=1 /
ijlg
? i, j = 1, 2, . . . , Njl ? 1, l
(37)
Nbi,j+1,l ≥ 2
STijl Qil VTjl
?i, j = 1, 2, ..., Njl ? 1, l
ijlg
≤ NbUP vtjlg ijl
ijlg g
? i, j = 1, 2, . . . , Njl ? 1, l, g ?i, j = 1, ..., Njl ? 1, l
(38) (39)
Again, in order to avoid non-linearities a set of available discrete sizes for the tank allocated after stage j, STFjl = {VTFjl1 , VTFjl2 , . . ., VTFjlGjl }, is selected. Let vt jlg be the binary variable that takes value 1 if a tank of size g is allocated in position j and zero otherwise. The ?rst tank size of the set, VTFjl1 , is equal to zero to represent “no tank allocation”. Then, Eqs. (26) and (27) are rewritten as: Nbijl ≥ 2
g=1 /
Nbijl =
4.3. Timing constraints Let tijl be the processing time for product i in stage j of plant l, TLil the cycle time of product i in plant l, and NTjl the number of groups of units operating out of phase in stage j of plant l where each group consist of NPjl units operating in phase. Therefore, TLil is determined using the following constraint: TLil ≥ tijl NTjl
STijl Qil VTFjlg
vtjlg ?i, j = 1, 2, . . . , Njl ? 1, l vtjlg ?i, j = 1, 2, ..., Njl ? 1, l
(28)
Nbi,j+1,l ≥ 2
g=1 /
STijl Qil VTFjlg
(29)
?i, j, l
(40)
vtjlg = exl
g
? j = 1, 2, . . . , Njl ? 1, l
(30)
The total time for producing i in plant l, Til , can be obtained using Til ≥ Nbijl TLil
?i, j, l
(41)
Eq. (30) states that if the plant l exists, then only one discrete size for a tank after stage j has to be selected. Using the continuous variable fijlg = Qil vtjlg , Eqs. (28) and (29) become linear Nbijl ≥ 2
g=1 /
All the products must be produced in the time horizon. Then Til ≤ Hl
i
l = 1, . . . , Nl
(42)
STijl VTFjlg
fijlg
?i, j = 1, 2, ..., Njl ? 1, l
(31)
The length of the time horizon of each plant, Hl , is a problem parameter. SPC is adopted as usual in this kind of models (Ravemark & Rippin, 1998).
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Introducing the binary variable xjln equal to 1 if n out of phase groups of units are used in stage j of plant l, and zero otherwise, then
NT UP
jl
Then, the equipment cost is
Nl Nj P NT UP NP UP
jl jl
EC = n xjln
ˇjl ?jl ndVFjlp l=1 j=1 p=1 n=1 d=1
jlpnd
NTjl =
n=1 NT UP
jl
?j, l
(43)
Nl
Nj
G ˇjl ?jl VTFjlg stjlg ? ?
+
l=1 j=1 g=1
(52)
xjln = exl
n=1
?j, l
(44)
For production plants and warehouses, a ?xed cost is considered if they are allocated. Then, the installation cost (LC) is given by
Nl Nm
UP where NTjl represent an upper bound for the number of units duplicated out of phase in stage j of plant l. Multiplying both sides of Eq. (40) by Nbijl , using Eq. (41) and the de?nition of NTjl given by Eq. (43), then: NT UP
jl
LC =
l=1
Cpll exl +
m=1
Cdepm ym
(53)
where Cpll and Cdepm are installation cost coef?cients. Therefore, the total investment cost is IC = Can (EC + LC) (54)
Til ≥
n=1
tijl n
xjln Nbijl
?i, j, l
(45)
In order to avoid non-linear constraints, a new variable is de?ned as wijln = Nbijl xjln , and replaced in Eq. (45). Considering NbUP ijl as an upper bound for the variable Nbijl , then
NT UP
jl
where the equipment and installation costs are annualized through a capital charge factor Can . 4.4.2. Operating cost The raw material cost, warehouse operating cost, and production cost are considered through the following expression
Ns Nr Np Nl Np Nl Nm
Til ≥
n=1
tijl n
wijln
?i, j, l
(46) OC =
Crawsr Qsril +
s=1 r=1 i=1 l=1 Np i=1 l=1 m=1 Nl
Cdim Qilm
where wijln ≤ NbUP xjln ijl
NT UP
jl
?i, j, l, n
(47)
+
i=1 l=1
Cprodil Qil
(55)
Nbijl =
n=1
wijln
?i, j, l
(48)
with Crawsr , Cdim , and Cprodil are appropriate cost parameters of the model. The Q amounts are expressed in kg per time horizon, therefore the cost parameters are given in $/kg, and OC represents the total annual operating cost. 4.4.3. Logistic cost The raw material distribution from raw material sites to production plants, the product distribution from production plants to warehouses, and from warehouses to customers zones are represented through the following cost expression
Ns Nr Np Nl Np Nl Nm
4.4. Objective function The objective function is the minimization of the total annual cost given by installation, investment, production, operating, and logistic costs. Following the expression for each cost is posed. 4.4.1. Investment and installation cost The investment cost considers the batch units and storage tanks cost. In this work, they are selected from a set of available discrete sizes. Then, equipment cost (EC) is given by
Nl Nj
UP UP Pjl NTjl NPjl
TC =
s=1 r=1 i=1 l=1 Np Nm Nk
Ctrawsrl Qsril +
i=1 l=1 m=1
Ctpilm Qilm
+
ˇjl ?jl VFjlp vjlp nxjln d xxjld i=1 m=1 k=1
Ctdimk Qimk
(56)
EC =
l=1 j=1 p=1 n=1 d=1 Nl Nj Gjl
+
l=1 j=1 g=1
ˇjl ?jl VTFjlg vtjlg ?
?
(49)
Ctraw, Ctp, and Ctd are cost coef?cients that depend on the product transported and the covered distance. Therefore, the objective function considered in this work is Minimize IC + OC + TC
The ?rst term corresponds to the batch units cost while the second one to the storage tanks cost. In order to avoid non-linearities, the continuous variable jlpnd is de?ned equal to 1 if stage j of plant l has n out of phase units and d in phase units of size p, 0 otherwise. Its value is given by
jlpnd
5. Case study All the examples were implemented and solved in GAMS (Brooke, Kendrick, Meeraus, & Raman, 1998) in an Intel (R) Core2, 1.86 GHz. The code CPLEX was employed for solving the MILP problems. The number of continuous and binary variables and constraints strongly depend on the number of SC nodes as well as the different plant design options considered.
≥ vjlp + xjln + xxjld ? 2 ?j, l, p, n, d
jlpnd
(50) (51)
0≤
≤ 1 ?j, l, p, n, d
G. Corsano, J.M. Montagna / Computers and Chemical Engineering 35 (2011) 149–164 Table 7 Raw material costs for Example 1. Distribution cost ($ kg?1 ) L1 S1 S2 0.02 0.14 L2 0.1 0.12 L3 0.08 0.14 L4 0.06 0.02 L5 0.08 0.06 Procurement cost ($ kg?1 ) R1 0.02 0.02 R2 0.02 0.01 R3 0.01 0.02
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5.1. Example 1 Consider a SC with 2 raw material sites that provide 3 different raw materials to 5 production plants where 4 products are produced through 3 batch stages. For each batch stage, a set of 5 discrete sizes (300 l, 500 l, 750 l, 1000 l, and 1200 l) and the option of duplication up to 2 units in phase or out of phase is considered. Also, 3 different tank sizes are available: 3000 l, 5000 l, and 10,000 l. There are 3 possible warehouses and 3 customer zones. Table 7 shows the raw material procurement and distribution costs from the sites to the production plants. In Tables 8 and 9, batch plant parameters are presented, while the distribution costs between plants and warehouses are shown in Table 10. Product demands of each customer zone and the distri-
bution cost from warehouses to customer zones are presented in Table 11. The optimal solution of the proposed simultaneous approach consists of a SC with only one production plant (L4). The SC and plant designs are shown in Fig. 5. As it can be noted, the ?rst stage is duplicated out of phase and there is a storage tank between stages 2 and 3. Raw materials are supplied by Site 2 and two warehouses are selected (M1 and M3). The total cost is equal to $1,019,500. A detailed list of the costs is provided in Table 12. In this approach, all costs are confronted simultaneously and, as the investment cost is more signi?cant than the other costs, the solution tries to use as few as possible units (and plants) for accommodating all the required production. In this example, the solution selects plant L4 because investment cost is lower than for others. Then, raw material site S2 supplies the three raw materials that are necessary for producing the four products, since the distribution cost from S2 to L4 is lower than from the other sites. The products are delivered to two warehouses M1 and M3, which reach the lowest distribution and installation costs from L4 to warehouses and from warehouses to customer zones. In order to show the impact of the simultaneous assessment of plant and SC design, a different approach is considered. The sequen-
Table 8 Batch plants parameters of Example 1. Size factors J1 I1 I2 I3 I4 0.9 0.6 0.7 0.8 J2 0.6 0.5 0.5 0.6 J3 0.4 0.4 0.4 0.4 Operating times (h) J1 14 12 16 10 J2 5 6 8 4 J3 7 4 5 5 Raw material conversion factor R1 0.8 0.6 0.4 0.5 R2 0.5 0.8 0.5 0.5 R3 0.7 0.8 0.5 0.5 Production cost ($ kg?1 ) L1 0.12 0.08 0.12 0.14 L2 0.18 0.16 0.14 0.08 L3 0.12 0.06 0.14 0.14 L4 0.06 0.12 0.08 0.04 L5 0.12 0.1 0.14 0.12
Table 9 Batch plant investment cost parameters. Unit cost coef?cient ?jl (annualized) L1 J1 J2 J3 Tanks 1620 2160 1890 500 L2 2430 1620 2700 500 L3 1350 2160 1890 500 L4 1350 1620 1890 500 L5 1890 1890 2430 500 Unit exponent coef?cient ˇjl For all the plants 0.6 0.6 0.7 0.6
Fig. 5. Optimal SC and plant design for the simultaneous approach of Example 1 (amounts × 103 kg).
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Table 10 Distribution cost between plants and warehouses for Example 1 ($ kg?1 ). M1 I1 L1 L2 L3 L4 L5 0.1 0.2 0.2 0.05 0.2 I2 0.17 0.19 0.18 0.1 0.18 I3 0.05 0.25 0.25 0.2 0.25 I4 0.05 0.25 0.25 0.15 0.25 M2 I1 0.2 0.19 0.18 0.15 0.2 I2 0.1 0.18 0.15 0.11 0.15 I3 0.15 0.35 0.25 0.2 0.25 I4 0.15 0.35 0.25 0.2 0.25 M3 I1 0.23 0.18 0.15 0.1 0.15 I2 0.16 0.19 0.08 0.15 0.15 I3 0.11 0.15 0.15 0.15 0.08 I4 0.11 0.15 0.18 0.05 0.08
Table 11 Product demands and distribution cost from warehouses to customer zones for Example 1. Product demand (kg) Distribution cost ($ kg?1 ) I2 130,000 120,000 130,000 I3 150,000 150,000 150,000 I4 100,000 100,000 120,000 M1 0.08 0.07 0.06 M2 0.09 0.09 0.07 M3 0.09 0.08 0.05
I1 K1 K2 K3 150,000 100,000 115,000
Table 12 Detailed list of costs of Example 1 ($ year?1 ). Costs Investment Production Raw material Warehouse Distribution Total Simultaneous approach 466,400 116,300 44,800 117,000 275,000 1,019,500 Sequential approach 629,300 119,100 44,800 118,400 231,100 1,142,700 No design approach 778,500 119,900 44,800 118,400 231,700 1,293,300
tial approach solves the SC design in a ?rst step. Then, plant designs are determined for those plants previously selected in the SC. As it was mentioned in the motivating example, the objective function in the ?rst step of the sequential approach minimizes the plant and warehouses installation, operating, and logistic costs (Eqs. (53), (55), and (56)) in order to ?nd the SC design that meets the product demands. In the second step, the investment cost is incorporated in
the objective function and the optimal design of the selected plants is obtained. The optimal SC design consists of two raw material sites, two production plants (L1 and L4) and three warehouses. The distribution ?ows into the SC are shown in Fig. 6. Fixing the SC con?guration, the design for both plants is solved in a second step. The optimal plant designs are shown in Fig. 7. The total cost is equal
Fig. 6. SC design for the sequential approach of Example 1 (amounts × 103 kg).
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159
Fig. 7. Optimal plant design of the sequential approach for Example 1 (amounts × 103 kg).
to $1,142,700 which is 12.1% higher than the total cost of the simultaneous optimization approach. Table 12 compares cost items of both approaches. As it can be noted, the main difference is reached on investment cost, where this cost is 35% worse for the sequential approach. Besides, the incorporation of plant design to the model affects not only the investment cost but also the SC design, which is very dissimilar in both cases since different raw material sites, plants and warehouses are selected. For the sequential approach, the logistic cost is 16% lower than for the simultaneous approach, since at the ?rst step of the approach, this cost has more in?uence than operating and installation cost on the objective function. Then, the logistic cost is dominant in the decision on which plants and which
warehouses should be allocated. For that reason, investment cost obtained in the second step is higher in the sequential approach, since it must be adjusted to previous decisions. It is worth highlighting that the plant con?guration and the production requirement of each plant affect to the production rate of each product. Table 13 shows the production rate for each product in each plant, PRil , and the total time for producing each product in each plant, Til , for both approaches. In this example, for the simultaneous approach all products are produced in only one plant. Therefore, in order to accommodate the production through the time horizon, unit duplication for the time limiting stage (stage 1) is required. Then, the cycle time for each product is reduced and consequently the production rate for each of them is increased. On the
Fig. 8. Optimal SC and plant designs for Example 1 without considering unit duplication and tank allocation (amounts × 103 kg).
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Fig. 9. Optimal plant design of the simultaneous approach for Example 1 with higher logistic cost (amounts × 103 kg). Table 13 Production rates and times for each product in each plant for Example 1. Simultaneous approach I1, L4 PRil (kg h?1 ) Til (h) 178.5 2044 I2, L4 312.5 1216 I3, L4 182.9 2460 I4, L4 250 1280 Sequential approach I1, L4 92.25 3832 I2, L1 166.6 2280 I3, L1 107.14 4200 I4, L4 125 2560
other hand, in the sequential approach the production is distributed in two plants. In this case, it is not necessary reduce the cycle time of each product since two products are produced in each plant instead of four. Therefore no duplication is used and the production rate of each product in each plant is lower than the production rates of the simultaneous approach. This is another advantage of the simultaneous model where plant con?guration is decided jointly with SC assignment and operational decisions (such as which facilities to allocate and how much to produce of each product in each). Finally, in order to evaluate the potential of the proposed approach, the same example is solved for obtaining the optimal SC and plant design simultaneously without considering unit duplications and intermediate storage tanks allocation. The total cost in this case is equal to $1,293,300 which is 26.8% higher than the simultaneous approach. The third column of Table 12 details these costs. In this case, the SC consists of two production plants (L1 and L4) since the total required production cannot be reached in only one plant without duplication and intermediate storage tanks allocation. Therefore, the options considered for the plant design also affect the global result. Fig. 8 shows the SC and plant designs for this solution. Table 14 displays the number of variables and constraints of each approach and CPU resolution times. CPU resolution time is longer for the simultaneous approach due to the number of binary variables involved and the tradeoffs considered. As previously mentioned, the investment cost in Example 1 has more weight than other costs in the overall objective function and
thus, the simultaneous approach gives the best solution. If logistic cost is incremented so that this cost is dominant in the objective function, the difference between simultaneous and sequential approach solutions is minor. For example, suppose the distribution costs from raw material sites to plants and from plants to warehouses are increased by four times from those presented in Table 10, while all the other model parameters are kept equal to the previous example. In this case, the optimal solution of the simultaneous approach consists of two production plants, L1 and L4, and plant con?gurations and production amounts are shown in Fig. 9. The optimal SC design is displayed in Fig. 10, with a total annual cost equal to $1,606,000. The same problem is solved with the sequential approach. The optimal solution for the ?rst step is identical to Example 1 and is shown in Fig. 6. Therefore, the optimal solution for the second step gives the same plant con?gurations previously obtained and presented in Fig. 7. The total annual cost is equal to $1,614,200. It can be noted that for this case, where logistic cost is dominant, the solutions of both approaches are similar. The only difference is that product I4 is produced in both plants in the simultaneous approach, while in the sequential approach, I4 is produced only in plant L4. The economic solution is a bit better (0.5%) for the simultaneous approach. Thus, it is important to note that the solution will depend on the logistic, production and investment cost magnitudes. As it was captured in the previous examples, the simultaneous approach evaluates the tradeoffs between these costs. When the investment
Table 14 Computational results of Example 1. Simultaneous approach Sequential approach 1st step # constraints # binary variables # continuous variable CPU time (s) 2124 223 2082 37.6 189 28 277 0.1 2nd step 916 84 902 2.0 1419 103 1062 3.6 No design approach
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161
Fig. 10. SC design for the simultaneous approach of Example 1 with higher logistic cost (amounts × 103 kg).
cost is higher than other costs, the simultaneous approach will give a very different solution from that obtained by other approaches. But depending on the impact of different costs in the objective function, the solutions of other approaches can be closer to the solution of simultaneous approach.
5.2. Example 2 In order to evaluate the performance of the proposed model, a larger example is addressed. Consider six raw material sites (S1–S6), with three different raw materials (R1–R3) in each site, ?ve
Fig. 11. Example 2: optimal SC and plant design for the simultaneous approach.
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Table 15 Optimal product distribution from warehouses to customer zones for Example 2 with the simultaneous approach. Customer zones K1 I1 I2 I3 I4 M5 M5 M3 M3 K2 M5 M5 M3 M3 K3 M4 M5 M3 M3 K4 M5 M5 M3 M3 K5 M4 M4 M3 M3 K6 M3 M3 M3 M3 K7 M1 M1 M1 M1 K8 M1 M1 M1 M1 K9 M5 M5 M3 M3 K10 M5 M5 M3 M3 K11 M4 M4 M3 M3 K12 M4 M4 M4 M4 K13 M4 M4 M4 M4 K14 M4 M4 M4 M4 K15 M5 M5 M5 M5
Customer zones K16 I1 I2 I3 I4 M4 M5 M3 M3 K17 M5 M5 M3 M3 K18 M5 M5 M3 M3 K19 M4 M5 M3 M3 K20 M4 M4 M3 M3 K21 M3 M3 M3 M3 K22 M1 M1 M1 M1 K23 M1 M1 M1 M1 K24 M4 M5 M3/M5 M3 K25 M4 M5 M5 M3 K26 M4 M4 M3 M3 K27 M4 M4 M4 M4 K28 M4 M4 M4 M4 K29 M4 M4 M4 M4 K30 M5 M5 M5 M5
Customer zones K31 I1 I2 I3 I4 M5 M5 M5 M3 K32 M5 M5 M5 M3 K33 M5 M5 M5 M3 K34 M5 M5 M5 M3 K35 M4 M4 M3 M3 K36 M3 M3 M3 M3 K37 M1 M1 M1 M1 K38 M1 M1 M1 M1 K39 M5 M5 M5 M3 K40 M4 M4/M5 M5 M3 K41 M4 M4 M3 M3 K42 M4 M4 M4 M4 K43 M4 M4 M4 M4 K44 M4 M4 M4 M4 K45 M5 M5 M5 M5
Table 16 Detailed list of costs of Example 2 ($ year?1 ). Costs Investment Production Raw material Warehouse Distribution Total Simultaneous approach 785,980 263,600 135,290 193,650 547,780 1,926,300 Sequential approach 1,182,850 263,600 148,270 185,400 502,090 2,282,210
multiproduct batch plants (L1–L5) with three stages each, where four products (I1–I4) can be produced, ?ve warehouses (M1–M5) and 45 customer zones (K1–K45). The set of available batch units and storage tanks sizes are the same of the previous example. Unit duplication in and out of phase and tank allocation is considered. The model parameters are not shown for space reasons, but they are available for interested readers. The solution of the simultaneous approach gives a SC that involves four raw material sites: S1, S2, S3 and S6; two batch plants: L3 and L4; and four warehouses: M1 and M3–M5. The SC and batch
plant designs are shown in Fig. 11, where customer zones are omitted for space reasons. The product distribution to all customer zones is displayed in Table 15. The total cost amounts to $1,926,400. A detailed list of costs is shown in Table 16. As in the previous example, the sequential approach is solved in order to compare both approach solutions. The ?rst optimization problem is to design the SC. The optimal SC con?guration for this Example contains ?ve raw material sites: S1–S4 and S6; three batch production plants: L2–L4; and four warehouses: M1 and M3–M5, for supplying the 45 customer zones. Then, the batch plant design is optimized, the SC network obtained in the previous step being the input data. The optimal SC design is shown in Fig. 12, and Fig. 13 shows the optimal plant designs. The product distribution from warehouses to customer zones is displayed in Table 17; and the detailed list of costs is shown in Table 16. Again, the main cost difference is in the investment cost, where the sequential approach is 50.5% worse. On the other hand, distribution costs are lower in the sequential approach, being reduced 6.7% the cost from raw material sites to plants, 14% from plant to warehouses, and 6.5% from warehouses to customer zones.
Fig. 12. Example 2: optimal SC design of the sequential approach.
G. Corsano, J.M. Montagna / Computers and Chemical Engineering 35 (2011) 149–164 Table 17 Optimal product distribution from warehouses to customer zones for Example 2 with the sequential approach. Customer zones K1 I1 I2 I3 I4 M4 M5 M5 M5 K2 M4 M5 M5 M5 K3 M4 M4 M5 M5 K4 M4 M5 M5 M5 K5 M4 M4 M3 M3 K6 M3 M3 M3 M3 K7 M1 M1 M1 M1 K8 M1 M1 M1 M1 K9 M4 M4 M1 M5 K10 M1/M4 M5 M1 M5 K11 M3 M3 M3 M3 K12 M4 M4 M1 M5 K13 M4 M4 M1 M5 K14 M4 M4 M1 M5
163
K15 M1 M1 M1 M5
Customer zones K16 I1 I2 I3 I4 M4 M4 M5 M5 K17 M4 M5 M5 M5 K18 M4 M5 M5 M5 K19 M4 M5 M5 M5 K20 M4 M4 M3 M3 K21 M3 M3 M3 M3 K22 M1 M1 M1 M1 K23 M1 M1 M1 M1 K24 M4 M5 M1 M5 K25 M4 M4 M1 M5 K26 M3 M3 M3 M3 K27 M4 M4 M1 M5 K28 M4 M4 M1 M5 K29 M4 M4 M1 M5 K30 M1 M5 M1 M5
Customer zones K31 I1 I2 I3 I4 M4 M4 M5 M5 K32 M4 M5 M5 M5 K33 M4 M5 M5 M5 K34 M4 M5 M5 M5 K35 M4 M4 M3 M3 K36 M3 M3 M3 M3 K37 M1 M1 M1 M1 K38 M1 M1 M1 M1 K39 M4 M4 M1 M5 K40 M4 M4 M1 M5 K41 M3 M3 M3 M3 K42 M4 M4 M1 M5 K43 M4 M4 M1 M5 K44 M4 M4 M5 M5 K45 M1 M5 M1 M5
Fig. 13. Example 2: optimal plant design of the sequential approach.
6. Conclusions In this work, a novel approach for the integration of SC and batch plant design was proposed. Decisions regarding SC network as nodes selection, supplier selection, material ?ows among nodes and product distribution are together considered with multiprod-
uct batch plant design decisions in order to attain a more integrated perspective of the SC design problem. The problem was formulated as an MILP model, where the embedded plant design model is a reformulation of the traditional one in order to keep the linear condition of the model. Batch unit sizes are selected according to a set of available discrete sizes, as
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it is usually found in the commercial and industrial practice. Unit duplications in and out of phase and allocation of storage tanks were considered as options to optimize the plant performance. It is worth highlighting that the incorporation of plant design into the SC design model gives a better integration of the whole network. The obtained solutions of the simultaneous optimization differ notably from the sequential or hierarchical approaches, both in economic values and in SC structure. Considering plant design into SC model optimization allows the simultaneous assessment of tradeoffs between different optimization variables. These evaluations cannot be carried out when sequential approaches are considered, as usual in the literature. On the other hand, it is very dif?cult to assign a ?xed and predetermined value to the installation of the plant taking into account that very different costs and performances can be attained depending on the included units and their structure. Thus, the main contribution of this work is to demonstrate that the optimization of plant design has a high impact on the SC design when they are simultaneously approached. Numerical examples were presented and solved with the proposed approach and confronted with a sequential approach. The examples show different cases and advantages, but, in general, we can conclude that sequential approach prioritizes in the ?rst step the logistic costs while, in the second step, investment costs are attained subject to previous decisions about the SC design. The proposed approach considers simultaneously the tradeoffs among all the costs. This is obviously an approximate model. Several parameters should be considered as uncertain factors in order to address a more realistic problem. However, this kind of formulation would signi?cantly increase the complexity of the problem. The proposed approach is the ?rst contribution where the impact of the design of the production facilities has been explicitly taken into account. Future works will extend this approach including uncertainty features in the formulation as well as ?exibility studies to attain more complete representations. Acknowledgments The authors are grateful for ?nancial support from CONICET and ANPCyT. References
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