Optimal Design of Multiechelon Supply Chain Networks with

Article pubs.acs.org/IECR

Optimal Design of Multiechelon Supply Chain Networks with Generalized Production and Warehousing Nodes Magdalini A. Kalaitzidou,† Pantelis Longinidis,‡ Panagiotis Tsiakis,§ and Michael C. Georgiadis*,† †

Department of Chemical Engineering, Aristotle University of Thessaloniki, University Campus, 54124 Thessaloniki, Greece Department of Engineering Informatics & Telecommunications, University of Western Macedonia, Karamanli & Lygeris Street, 50100 Kozani, Greece § Wipro Consulting Services, 3 Sheldon Square, London W2 6PS, United Kingdom ‡

S Supporting Information *

ABSTRACT: This work proposes a mathematical modeling framework for the design of supply chain networks by providing flexibility on facilities’ location and operation. The proposed model addresses the design of a multiproduct and multiechelon network consisting of generalized production/warehousing nodes that can receive any material from any potential supplier or any other generalized production/warehousing node and deliver any material to any market or any other generalized production/ warehousing node. The model is formulated as a mixed integer linear programming problem that aims to find the optimal structure of the network in order to satisfy market demand with the minimum overall capital and operational cost. The applicability of the proposed generalized supply chain network design model is illustrated by using a real case study from a European consumer goods company whereas its robustness and value are documented through sensitivity analysis and through a comparison with a counterpart model that utilizes the mainstream fixed echelon network structure.

1. INTRODUCTION During the past two decades, supply chain management (SCM) has become a popular research stream among academics and

Figure 2. Generalized supply chain network structure.

channels, and how to organize the interfaces among the various parties in the supply chain.9 On the other hand, SCO is the process of determining solutions to more tactical issues such as local inventory polices and deployment, manufacturing and service schedules, and transportation plans, etc.10 SCND is catalytic in ensuring effective SCM as its fundamental decisions determine the configuration of the supply chain and pose constraints within which SCO takes place. Decisions concerning facilities’ location, role, and performed processes affect the flexibility that the supply chain has in changing the way it meets demand while capacity allocation decisions are intertwined with assets utilization,

Figure 1. Mainstream supply chain network structure.

practitioners as its interface with organizational effectiveness, corporate performance, and competitive advantage is documented not only in scientific literature1−6 but also in industry reports7 as well as in trade journals.8 The abundant and growing body of literature about SCM, within a broad spectrum of scientific domains, develops decision support models that tackle problems arising in the two core phases of SCM, namely, supply chain network design (SCND) and supply chain operation (SCO). SCND involves strategic decisions and plans regarding where to locate facilities (for production, storage, distribution, and retail, etc.), how to allocate capacities or assign production tasks to the various facilities, how to choose and develop supplier and distribution © 2014 American Chemical Society

Received: Revised: Accepted: Published: 13125

May 12, 2014 July 22, 2014 July 29, 2014 July 29, 2014 dx.doi.org/10.1021/ie501931n | Ind. Eng. Chem. Res. 2014, 53, 13125−13138

Industrial & Engineering Chemistry Research

Article

Table 1. Structure Features of Relevant and Recent SCND Modelsa echelons article

S

P

Sabri and Beamon22 Tsiakis et al.23 Talluri and Baker24,b Chen et al.25,c Yan et al.26 Lababidi et al.27 Miranda and Garrido28 Eskigun et al.29 Guillén et al.30 Santoso et al.31,d Guillén et al.32 Melo et al.33 ́ et al.34 Lainez Romeijn et al.35,f Azaron et al.36 Manzini et al.37 Naraharisetti et al.38 Selim and Ozkarahan39 Thanh et al.40 Tsiakis and Papageorgiou41 You and Grossmann42 You and Grossmann43 Xu et al.44 Naraharisetti et al.45 Altiparmak46 Gümüs et al.47,b Hammami et al.48 Mohammadi Bidhandi et al.49 ́ et al.50 Lainez Costa et al.51 Franca et al.52 Lee et al.53 Park et al.54,f You and Grossmann55 Cardona-Valdés et al.56 Georgiadis et al.57

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Longinidis and Georgiadis58 Mohammadi Bidhandi and Mohd Yusuff59 Shukla et al.60 You and Grossmann61 Melo et al.62 Olivares-Benitez et al.63 Akgul et al.64,g Akgul et al.65,g Pishvaee et al.66 Sadjady and Davoudpour67,f Baghalian et al.68 Bassett and Gardner69 Cardoso et al.70 Hammami and Frein71 Latha Shankar et al.72 Liu and Papageorgiou73,h Longinidis and Georgiadis74 Tabrizi and Razmi75 Fumero et al.76 Longinidis and Georgiadis77 Rodriguez et al.78 Yongheng et al.79 our model

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Abbreviations: S, supplier; P, plant; W, warehouse; DC, distribution center; M, market; BE, between echelons; WE, within echelons; PD, predetermined; OP, optimized. bDistributors are the demand points. c Markets echelon consists of retailers and customers. dPlants echelon consists of manufacturing centers and finishing facilities. eIndistinguishable nodes. fRetailers are the demand points. gSuppliers echelon consists of biomass cultivation sites. hPlants echelon consists of active ingredients echelon and formulation plants echelon. iP/W hybrid nodes.

• • • • • • • •

models, which provide optimal solutions within a given set of degrees of freedom, deluged academic journals. The number of review papers on SCND confirms this assertion and indicates the fact that this research branch is flourishing hitherto. The early reviews in SCND literature deal with models that optimize production, distribution, and inventory systems in an integrated manner, rather than sequentially, within the overall supply chain. Vidal and Goetschalckx12 review eight strategic production−distribution mixed integer programming (MIP) models with an emphasis on global supply chains while Sarmiento and Nagi13 study 16 production−distribution models that considered transportation system explicitly. By moving from infancy to conceptual development, SCND gained further popularity that was mirrored by more reviews. On these lines, Meixell and Gargeya14 survey 18 global supply chain design models, Shah15 describes the state of the art in process industry supply chains, Akçalı et al.16 conduct an annotated bibliography on 22 network design models for reverse and closed-loop supply chains, Papageorgiou17 reviews

responsiveness, and costs realized to satisfy customer demand.11 Locating plants, warehouses, distribution centers, and retail outlets are long-term investments that cannot be altered on short notice without financial burdens. Likewise, assigning processes and capacities to each facility requires capital investments in technological equipment, in warehousing management systems, and in transportation infrastructure. Therefore, decision making in SCND engages huge amounts of funds, shackled in non-earning assets and processes, with capital recovery projected in the long run. Undoubtedly, there is no room for “common sense” and “business as usual” in SCND as these approaches, sooner or later, will blindly lead to suboptimal, or even infeasible, SCO plans and consequently to SCM failure. SCND is the initial pillar of SCM’s commitment in creating added value with less cost and, as such, deserves meticulous and perpetual consideration. This challenge has motivated researchers to build optimization models, supporting decision makers in SCND, and the results have been impressive. A wealth of 13126

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nodes without any capability to define the optimal operation of each node. This work aims at enriching the literature on SCND by introducing a general mathematical programming framework that utilizes an innovative composition for the supply chain network. This framework integrates the different components of a supply chain without any a priori assumption on the fundamental structure of the network. The generalized production/warehousing (P/W) nodes substitute the traditional plants, warehouses, and distribution centers while their function is not known in advance but optimally defined by the optimization procedure. These generalized nodes along with suppliers and markets form the generalized supply chain network which is illustrated in Figure 2. The rest of the work is structured as follows. Section 2 reviews the most relevant and recent works in SCND literature. Section 3 introduces the generalized SCND problem and presents its mathematical formulation. The applicability of the proposed model is illustrated, through a real case study, in section 4 followed by concluding remarks, managerial implications, and further research directions.

62 strategic and tactical supply chain optimization models for the process industry, Melo et al.18 analyze 98 facility location models in the context of SCM, Nikolopoulou and Ierapetritou19 studied seven sustainable supply chain design models, Lambiase et al.20 review 50 supply chain design and strategic planning models, and Farahani et al.21 survey 50 SCND models and 39 competitive SCND models. Extant SCND models studied within the aforementioned reviews and those currently populating the operations research/ management science (OR/MS) literature assume a fundamental structure of the network, as shown in Figure 1, with distinct and consecutive echelons. Several nodes with predetermined function constitute each echelon, and the product flow moves from an echelon’s node to a subsequent echelon’s node in a downstream fashion. The first echelon contains suppliers, the second manufacturing plants and/or assembling factories, the third warehouses, the fourth distribution centers and/or retail outlets, and finally the fifth contains markets and/or customer zones. In some models several echelons are ignored, mainly suppliers and warehouses, while in some others additional echelons are introduced. The latter applies solely to closed loop supply chains where collection centers and remanufacturing plants exist by default. In each of these echelons, with the exception of the first and the last, a set of potential alternative locations are considered for facilities establishment. In the first echelon there is a set of established potential suppliers to be selected for sourcing whereas in the last echelon there is no selection process as all markets should be served. By selecting the optimal nodes and locations within each echelon as well as their optimal connection links, a supply chain network is formed which has the function of moving material objects from sources of production to end customers. The flow of materials, intermediate products, and end products strictly follows the pattern depicted in Figure 1 where the function of each node is determined from the echelon it belongs to rather than defined optimally. This structure deprives flexibility from the supply chain network as at least one node from each echelon should be selected in order to fulfill customer demand. To wit, at least one supplier should be contracted with at least one plant in order to receive raw materials and transform them to end products. These end products should be sent to at least one warehouse for storage and then should be shipped to at least one distribution center which in turn distributes them to markets. Even if some intermediate nodes are nearly substitute−mainly warehouses and distribution centers as their operation is quite similar at strategic level−they are selected in order to form the supply chain network. For instance, it is not possible for a warehouse (or a plant) to supply a market directly, without passing through the distribution center echelon. In fact, in many cases the distinction between “warehouse” and “distribution center” is somewhat arbitrary, and insisting upon having two separate stages of material storage throughout the supply chain network may lead to uneconomical decisions. On the other hand, in some cases, especially when extended geographical distances are concerned, it may be advantageous to establish three or more stages of warehousing between production plants and markets. This would allow the exploitation of economies of scale in transportation as a higher proportion of the material flow shares common routes over longer distances. Moreover, this structure allows operations to take place only in specific

2. LITERATURE REVIEW As mentioned earlier, the literature on SCND is vast due to its catalytic role in effective SCM. An exhaustive literature review is beyond the scope of this work as many excellent review papers exist. Thus, our literature review aims to present the supply chain network structures adopted in relevant SCND models and to highlight the novelty of our research endeavor. On these lines, Table 1 presents the structural features of relevant SCND models and more specific the echelons involved, the flow between echelons and within echelons, and the approach with which the role of a node is decided. One can first observe from Table 1 that plants echelon, distribution centers echelon, and markets echelon are the most popular levels. This is reasonable as, in the context of SCM, without production, distribution, and market demand we cannot speak for a supply chain system. On the other hand, suppliers and warehouses are entities less popular but necessary to comply with the trend toward integrating all entities participating in downstream and upstream relationships. The vast majority of SCND models restricts the flow of material from echelon to echelon and not between nodes included in the same echelon. This interlayer flow is very important for capacity allocation. Finally, the function of each node is defined by the echelon it belongs to and is selected from a set of potential alternatives. ́ et al.,50 none of the With the exception of the work of Lainez existing models allow the optimization procedure to decide the role of each node independently of its echelon and more importantly none of existing models allow a node to have combined capabilities and thus constitute a hybrid node. The present work aims to fill this gap and propose a generic SCND framework that allows these hybrid nodes to exist. The work of ́ et al.50 considered a modeling framework that resembles Lainez ours with allowed flows within facilities in which equipment and tasks are determined by the optimization procedure. Their mathematical formulation employed the state task network (STN) representation80 while their model required dummy variables to address the case where suppliers and markets are located in the same areas with indistinguishable nodes. 13127



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If a node n ∈ PW is established (Yn = 1), it should receive material from at least one other noden′ ∈ S ∪ PW\{n} and should provide material to at least one other node n′ ∈ C ∪ PW \{n}. As shown in constraint (3), if a node n ∈ PW is established (Yn = 1), the binary variable that expresses the establishment of a material transportation link (Xn′n) is forced to take the value one for at least one pair of n′ ∈ S ∪ PW with n ∈ PW and provided that n ≠ n′. In the same manner, constraint (4) shows that if a node n ∈ PW is established (Yn = 1), the binary variable that expresses the establishment of a material transportation link (Xnn′) is forced to take the value one for at least one pair of n ∈ PW with n′ ∈ PW ∪ C, provided that n ≠ n′.

3. PROBLEM DEFINITION AND MATHEMATICAL FORMULATION 3.1. Problem Statement. This work addresses the design of a multiproduct and multiechelon supply chain network consisting of a set of spatially distributed markets and suppliers, whose location is known, and a set of facilities, whose locations are to be determined, to satisfy customer demands. An innovative configuration to the network’s structure is introduced by entering a level consisting of generalized P/W nodes whose function is not a priori assumed, as in mainstream fixed echelon SCNs. The network’s structure and the P/W’s function are left to be determined by the optimization procedure. Generalized P/W nodes can have either production capability or warehousing capability or both. Moreover, they receive material from any potential supplier or any other P/W node and deliver material to any market or any other P/W node, as shown in Figure 2. Another significant characteristic of the generalized network is the collaboration among P/W nodes. The model permits intralayer material flow connections that occur in the P/W level. All products can be produced using a varied amount of manufacturing resources, as long as P/W node takes the role of manufacturing plant by the optimization procedure. Certain constraints are applied to the structure of the network, to the flow of materials within the network, to the core operations in the network (purchasing, production, and warehousing), and to customer satisfaction. The objective is to minimize the overall capital and operational cost of the network and to determine the optimal structure of the network. The proposed SCND model yields exact solutions for the following decisions: (a) the number of contracted suppliers, (b) the number, location and role of generalized P/W nodes, (c) the material flow among the supply chain network’s levels, and (d) the functional elements (capacity, material flow, and rate of operation). 3.2. Mathematical Model. A deterministic MILP model is formulated where each product can be produced at several P/ W nodes in different locations and product demand is assumed to be known and time-invariant. Moreover, all transportation flows determined by the optimization are considered to be time-averaged quantities. We denote the set of all nodes in the network as n ∈ N. This includes not only the generalized nodes n ∈ PW but also suppliers nodes n ∈ S and market nodes n ∈ C. Overall we have N = S ∪ PW ∪ C. The MILP optimization problem consists of a cost minimization objective function (OBJ) and constraints for network structure, transportation flows, production resources, material supply, product demand, and warehouse capacity. 3.2.1. Network Structure Constraints. Constraints (1) and (2) demonstrate the conditions for the establishment of a node n ∈ PW. In specific, constraint (1) states that if a production capability is established at a node n ∈ PW (YPn = 1) then the corresponding node n ∈ PW should be established as the binary variable that expresses its establishment is forced to take the value of one (Yn = 1). In the same fashion, constraint (2) states that if a warehousing capability is established at node n ∈ PW (YW n = 1), then the corresponding node n ∈ PW should be established as the binary variable that expresses its establishment is forced to take the value of one (Yn = 1). Yn ≥ YnP ,

∀ n ∈ PW

Yn ≥ YnW ,

∀ n ∈ PW

∑

Yn ≤

Xn′n ,

∀ n ∈ PW (3)

n ′∈ S ∪ PW\{n}

∑

Yn ≤

X nn ′ ,

∀ n ∈ PW

n ′∈ C ∪ PW\{n}

(4)

A connection between a node n′ ∈ S and a node n ∈ PW can exist only if both the supplier is contracted and the generalized node is established. Constraint (5) forces the binary variable expressing the contracting of node n′ ∈ S(YSn′) to be unity when the material transportation link, between a node n′ ∈ S and a node n ∈ PW, is established(Xn′n = 1). On the other hand, constraint (6) forces the binary variable expressing the establishment of node n ∈ PW to be unity when the material transportation link, between a node n′ ∈ S and a node n ∈ PW, is established (Xn′n = 1). X n ′ n ≤ Y nS′ , X n ′ n ≤ Yn ,

∀ n′ ∈ S , n ∈ PW, n ≠ n′ ∀ n′ ∈ S , n ∈ PW, n ≠ n′

(5) (6)

Similarly, a connection between two nodes n ∈ PW and n′ ∈ PW can exist only if both nodes are established. Constraints (7) and (8) stress this condition while constraint (9) requires the establishment of node n ∈ PW if it is going to transfer material to node n′ ∈ C. X nn ′ ≤ Yn ,

∀ n ∈ PW, n′ ∈ PW, n ≠ n′

(7)

X nn ′ ≤ Yn ′ ,

∀ n ∈ PW, n′ ∈ PW, n ≠ n′

(8)

X nn ′ ≤ Yn ,

∀ n ∈ PW, n′ ∈ C , n ≠ n′

(9)

The binary variable (Xnn′) is fixed to zero, as the model does not allow reverse flows (constraints (12) and (13)), intralayer flows between suppliers (constraint (10)) and markets (constraint (11)), and direct flows from suppliers to markets (constraint (14)). X nn ′ = 0,

∀ n ∈ S , n′ ∈ S , n ≠ n′

(10)

X nn ′ = 0,

∀ n ∈ C , n′ ∈ C , n ≠ n′

(11)

X nn ′ = 0,

∀ n ∈ C , n′ ∉ 0, n ≠ n′

(12)

X nn ′ = 0,

∀ n ∈ PW, n′ ∈ S , n ≠ n′

(13)

X nn ′ = 0,

∀ n ∈ S , n′ ∈ C , n ≠ n′

(14)

3.2.2. Transportation Flow Constraints. The flow (Qinn′) of material i from node n ∈ S to node n′ ∈ PW is limitied between upper and lower bounds provided that the corresponding transportation connection has been established:

(1) (2) 13128


Industrial & Engineering Chemistry Research max Q inn ′ ≤ Q inn X , ′ nn ′

∀ i , n ∈ S , n′ ∈ PW, n ≠ n′

Article

3.2.5. Product Demand Constraint. The total rate of each material i arriving at market node n′ ∈ C from all nodes n ∈ PW must be equal to market demand:

(15)

Similarly, the flow of material i from node n ∈ PW to node n′ ∈ PW and from node n ∈ PW to node n′∈C can exist, only if the corresponding connections is established. Also the flows must be limited by given upper bounds: Q inn ′ ≤

max Q inn X , ′ nn ′

Q inn ′ ≤

max Q inn X , ′ nn ′

∑

(16) (17)

WnminYnW ≤ Wn ≤ WnmaxYnW ,

Moreover, when there is connection from node n∈N to node n′∈N, there is a minimum total material flow that is needed to justify the link: min Q nn X ≤ ′ nn ′

∑ Q inn′,

∀ n ∈ S , n′ ∈ PW, n ≠ n′ (18)

∑ Q inn′,

∀ n ∈ PW, n′ ∈ PW, n ≠ n′

Wn ≥

i

∑ Q inn′,

(20)

Q in ′ n +

∑

vikξkn =

k∈ Kn

∑

Q inn ′,

∀ i , n ∈ PW

n ′∈ C ∪ PW\{n}

(21)

The term (vik) expresses the amount of material i produced by the unit amount of operation taskk. Moreover the rate of production is given by multiplying the term vik with the continuous variable that expresses the rate of operation of a task k at node n ∈ PW(ξkn). The total utilization of each resource e is limited to the total rate of availability of manufacturing resource e at node n ∈ PW(Een) as shown in constraint (22).

∑

λek ξkn ≤ Een ,

∀ e , n ∈ PW (22)

k ∈ Kn

Provided that node n ∈ PW has the production capability established, the total rate of availability of manufacturing resource e at node ∈ PW(Een) will be between lower and upper bounds: EenminYnP ≤ Een ≤ Eenmax YnP ,

∀ e , n ∈ PW

(23)

3.2.4. Material Supply Constraints. Upper and lower bound are imposed also for the purchased amounts of material i from the selected node n ∈ S, provided that the corresponding supplier has been contracted. SinminY nS ≤ Sin ≤ SinmaxY nS ,

∀ i, n ∈ S

(24)

The total amount of material i provided and transferred by a supplier node n ∈ S to node n′ ∈ PW is given by constraint (25):

∑ n ′∈ PW

Q inn ′ = Sin ,

∀ n ∈ PW

(27)

∑

Q inn ′ainoutbound ,

i , n ′∈ C ∪ PW

(28)

3.2.7. Objective Function. The objective is to minimize the overall capital and operational cost of the network. Capital cost consists of infrastructure cost whereas materials’ handling, production, transportation, and purchasing contribute to operational cost. Infrastructure cost is related to the establishment of a warehouse capability or a production capability at a particular node n ∈ PW. If a production capability is established at a node n ∈ PW, then its infrastructure cost has a fixed element (CpnYpn) and a variable element (∑eEenγPen). The former is the annualized fixed cost required to establish a production capability (CPn ). The latter is the summary of the products of the availability rate of resource e(Een) and the unit cost of this resource (γPen). Similarly, warehouse infrastructure cost has a fixed element which is the annualized fixed cost required to establish a warehousing capability (CW n ), and a variable element which is the product of the unit warehousing cost (γW n ) and the warehousing capacity (Wn). The sum of the preceding constitutes the total infrastructure cost. Regarding operational cost, handling cost is expressed as a linear function of the total throughput at node n ∈ PW. The total throughput is calculated by summarizing the summary of the continuous variables expressing the rate of flow of material i that arrives at node n ∈ PW from node n ∈ S or/and from the other node n ∈ PW and the summary of the continuous variables expressing the rate of flow of material i that leaves node n ∈ PW to node n ∈ C or/and to other noden ∈ PW. By multiplying the aforementioned total throughputs with the unit handling cost for material i(CinWH) and summarizing the resulting products, we obtain the handling cost. Production cost is related to the utilization of various resources e at noden ∈ PW and is determined as the summary of the products of the unit cost of consumption of resource e at node n ∈ PW(δPen) with the total utilization of each resource e. Utilization is the product of the amount of manufacturing resource e required to perform the unit amount of operation of task k(λek) and the rate of operation of task kat node ∈ PW(ξkn). Transportation cost is decomposed into three terms. The first term expresses the transportation cost of material i transferred from node n′ ∈ S to node n ∈ PW, the second term

3.2.3. Production Resources Constraints. In nodes n ∈ PW where production capability is established the overall balance for the production of material i is the inflow minus the outflow of material i plus the rate of production of material i at that node, as shown in constraint (21).

∑

Q in ′ naininbound +

∀ n ∈ PW, n ≠ n′

∀ n ∈ PW, n′ ∈ C , n ≠ n′

i

n ′∈ S ∪ PW\{n}

∑ i , n ′∈ S ∪ PW

(19) min Q nn X ≤ ′ nn ′

(26)

Provided that a warehousing capability is established, the warehouse capacity is approached as a linear function of the handled material flow as shown in constraint (28) with (ainbound )/(aoutbound ) expressing the relationship between the in in capacity of the warehouse at node n ∈ PW and the material i handled that enters/leaves the node.

i

min Q nn X ≤ ′ nn ′

∀ i , n′ ∈ C

3.2.6. Warehouse Capacity Constraints. If a warehouse capability is established at node n ∈ PW, it should be between lower and upper bounds:

∀ i , n ∈ PW, n′ ∈ PW, n ≠ n′

∀ i , n ∈ PW, n′ ∈ C , n ≠ n′

Q inn ′ = Din ′ ,

n ∈ PW

∀ i, n ∈ S (25) 13129



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Figure 3. Potential nodes of the case study network.

4. CASE STUDY The proposed modeling framework is applied in a modified version of a real case study within a European consumer goods

expresses the transportation cost of material i transferred from node n ∈ PW to the other node n′ ∈ PW, and the third term expresses the transportation cost of material i transferred from node n ∈ PW to node n′ ∈ C. By summarizing the previous three terms for all material i, we have the transportation cost. Finally, purchasing cost has a fixed element (CSnYSn), the cost required to establish a relationship (supplier management costs, technology integration investments, and hedging costs for financial risk, etc.) with node n ∈ S (CSn), and a variable element (CSinSin), the product of the unit purchase price of material i from node n ∈ S (CSin) and the continuous variable expressing the purchased amounts of material i from the selected node n ∈ S (Sin). By summarizing this variable element for all materials, we obtain the purchasing cost from each node n ∈ S, and then by adding the resulting variable element with the fixed element and summarizing for all suppliers, we obtain the purchasing cost. The sum of the preceding cost elements constitutes the objective function (29):

Table 2. Annualized Fixed Cost (CSn) of establishing Relationship with Each Supplier (rmu/year) supplier

CSn

RU UA TR BG RO

9,000 13,000 7,000 11,000 7,000

Table 3. Unit Purchase Price (CSin) of Each Material from Each Supplier (rmu/ton) CSin material i i i i i i i i i i i i i i

13130

= = = = = = = = = = = = = =

1 2 3 4 5 6 7 8 9 10 11 12 13 14

RU

UA

TR

BG

RO

1.25 2.25 3.25 1.25 1.25 2.25 1.25 1.75 3.25 3.25 4.33 3.25 1.25 1.65

2.55 1.55 3.55 1.55 1.55 1.46 1.31 2.33 1.46 2.55 1.63 1.55 1.55 1.55

1.98 2.98 3.98 1.98 2.98 3.98 1.98 1.98 1.98 1.98 1.91 1.55 3.91 2.98

2.93 2.93 1.93 2.93 1.93 1.33 1.93 1.03 1.93 2.93 1.93 1.93 1.93 2.01

1.67 1.55 1.85 1.85 2.85 3.85 1.85 1.85 3.85 2.85 2.85 3.05 1.85 2.85



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Table 4. Maximum Rate of Availability (Emax en ) of Each Resource in Each P/W (hours/year) Emax en resource e e e e e e

= = = = = =

1 2 3 4 5 6

UK

ES

IT

FR

SE

IE

NL

GR

DK

FI

PT

BE

CH

NO

AT

228 245 251 231 251 231

223 233 260 236 260 236

228 226 234 239 234 239

235 241 222 257 236 242

265 217 258 242 236 218

281 211 212 222 237 215

217 258 252 244 229 211

232 231 218 251 246 232

218 238 248 258 268 278

211 261 256 247 237 218

213 228 249 252 239 236

218 278 268 235 225 225

252 236 249 275 285 236

252 238 248 242 246 235

248 272 236 249 252 248

Table 5. Establishment Cost (γPen) for Each Unit of Manufacturing Resource at Each P/W(rmu/hour) γPen resource e e e e e e

= = = = = =

1 2 3 4 5 6

UK

ES

IT

FR

SE

IE

NL

GR

DK

FI

PT

BE

CH

NO

AT

0.20 0.32 0.36 0.22 0.36 0.22

0.14 0.23 0.40 0.24 0.41 0.24

0.20 0.17 0.21 0.25 0.21 0.25

0.25 0.29 0.15 0.40 0.25 0.29

0.46 0.12 0.41 0.29 0.25 0.13

0.57 0.08 0.08 0.15 0.26 0.11

0.12 0.41 0.36 0.31 0.20 0.08

0.22 0.22 0.13 0.36 0.32 0.22

0.13 0.27 0.34 0.41 0.48 0.55

0.08 0.43 0.39 0.33 0.26 0.13

0.09 0.20 0.34 0.36 0.27 0.25

0.13 0.51 0.44 0.24 0.17 0.17

0.36 0.25 0.34 0.53 0.60 0.25

0.36 0.27 0.34 0.29 0.32 0.25

0.34 0.50 0.25 0.34 0.36 0.34

Table 6. Annualized Fixed Cost (Cpn) of Establishing Production Capability at Each P/W (rmu/year) P/W

Cpn

UK ES IT FR SE IE NL GR DK FI PT BE CH NO AT

40,000 15,000 10,000 20,000 22,500 10,000 20,000 10,000 15,000 5,000 12,500 25,000 35,000 17,500 32,500

Table 8. Amount of Each Resource Required to Perform Unit Amount of Each Task (λek, hour/task) λek resource e e e e e e

νki i i i i i i i i i i i i i i

= = = = = = = = = = = = = =

1 2 3 4 5 6 7 8 9 10 11 12 13 14

k=1

k=2

k=3

k=4

0.33 0.25 0.35 0.25 0.25 0.25 0.25 0.25 0.25 0.25 0.25 0.25 0.25 0.25

0.55 0.55 0.55 0.46 0.32 0.55 0.67 0.55 0.55 0.55 0.55 0.55 0.55 0.55

0.98 0.98 0.46 0.98 0.38 0.98 0.58 0.98 0.98 0.98 0.98 0.98 0.98 0.98

0.93 0.93 0.93 0.93 0.93 0.93 0.88 0.32 0.55 0.93 0.93 0.58 0.93 0.93

1 2 3 4 5 6

k=1

k=2

k=3

k=4

0.033 0.045 0.051 0.032 0.052 0.030

0.023 0.033 0.060 0.036 0.060 0.036

0.028 0.026 0.034 0.039 0.034 0.039

0.035 0.040 0.022 0.057 0.036 0.042

company that we have experienced for several years.23,57 All of the necessary data were supplied by the company, but since they are treated as confidential they have been scaled with a common factor while real currency units have been substituted with relative money units (rmu). The company during its business process re-engineering program is looking for an innovative way to configure its European supply chain network so as to save costs. The potential supply chain network is comprised of 38 nodes where five of them concern supplier nodes, 15 of them concern P/W nodes, and 18 of them concern market nodes, as shown in Figure 3.The potential suppliers’ locations are in Russia (RU), Ukraine (UA), Turkey (TR), Bulgaria (BG), and Romania (RO). All suppliers can provide the full range of 14 materials to the P/W nodes. Table 2 presents the annualized fixed cost of establishing a relationship with a supplier and Table 3 the unit purchase price of material from a supplier. The maximum rate of availability of each material by each supplier is 10,000 tons/ year while the minimum is 10 tons/year. The company has 15 potential locations for the P/W nodes, and these are in United Kingdom (UK), Spain (ES), Italy (IT), France (FR), Sweden (SE), Ireland (IE), The Netherlands (NL), Greece (GR), Denmark (DK), Finland (FI), Portugal (PT), Belgium (BE), Switzerland (CH), Norway (NO), and Austria (AT). Table 4 presents the maximum rate of availability of each manufacturing resource in each P/W node, while the minimum is 5 h/year for all resources. Table 5 presents the cost to install a unit of manufacturing resource at each P/W, while

Table 7. Amount of Each Material Produced by a Unit Amount of Each Task (νik, ton/task) material

= = = = = =

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Table 9. Unit Cost (δPen) of Consumption of Each Manufacturing Resource at Each P/W (rmu/hour) δPen resource e e e e e e

= = = = = =

1 2 3 4 5 6

UK

ES

IT

FR

SE

IE

NL

GR

DK

FI

PT

BE

CH

NO

AT

22 24 55 23 35 53

52 23 36 13 26 53

62 22 33 33 33 43

53 54 32 25 33 54

36 41 55 54 43 31

58 51 41 32 43 31

37 58 52 44 29 41

62 31 58 41 36 52

48 48 48 55 66 47

41 41 56 34 47 58

61 52 64 35 33 33

28 28 28 15 25 35

35 43 44 47 38 23

52 38 38 52 46 35

48 72 36 49 52 48

Figure 4. Optimal generalized supply chain network of the case study.

characterized by a unit cost of consumption of the resource in each P/W, as shown in Table 9. The warehousing capacity of each P/W is approached as a linear function of handled material flow. The coefficients relating the capacity to the throughput of each material entering/leaving each node are ainbound = 1 and aoutbound = 1.1, in in respectively. In some occasions when there are special standards or demands from the customers, the products entering the warehouse may go through repackaging; therefore we assume aoutbound > ainbound . Supporting data for the unit in in handling cost of all materials, the fixed cost for establishing warehousing capability, the establishment cost for each unit of warehousing capability, and the maximum warehousing capacity at each P/W are respectively represented in Table S1 and Table S2 in the Supporting Information material. The minimum warehousing capacity is zero. The markets are in 18 different countries, United Kingdom (UK), Spain (ES), Italy (IT), France (FR), Sweden (SE), Ireland (IE), The Netherlands (NL), Greece (GR), Denmark (DK), Finland (FI), Portugal (PT), Belgium (BE), Switzerland (CH), Norway (NO), Austria (AT), Germany (DE), Poland (PL), and Turkey (TR). The company implements a single service policy for its markets. The rate of flow of each material that can be transferred from one node to another is preserved

Table 10. Optimal Total Flow of Material (Qin′n) from Supplier to P/W (tons/year) Qin′n supplier

ES

IT

BE

CH

BG RO

1317 1892

1520 2204

6051 11,237

10,724 10,186

Table 11. Optimal Total Flow of Material (Qinn′) from P/W to P/W (tons/year) Qinn′ P/W

BE

CH

ES T BE

190

898 1615 2133

Table 6 presents the cost to install production capability at each P/W. Each of the 14 materials is produced by four different tasks. The amount of each material produced by a unit amount of each task is shown in Table 7 while the amount of each manufacturing resource required to perform a unit amount of each task is given in Table 8. Every manufacturing resource is 13132


Industrial & Engineering Chemistry Research 5,550 6,074 6,167 6,450 6,746 6,783 7,095 7,421 3,643 3,807

DK

3,987

GR

4,046

NL

4,230

IE

4,426

SE 32,453

FR IT

32,867

ES UK

32,653 ES IT BE CH

P/W

among a lower bound of 100 tons/year and an upper bound of 5,000 tons/year. Additionally, the total rate of material flow, whatever type, from one node to another is restricted between 100 and 10,500 tons/year. Finally, in the case study applied in our model, transportation costs are independent of material type. Appropriate data for the demand for each material from each market and transportation costs are provided in Tables S3−S6 in the Supporting Information material. 4.1. Results. The proposed model was solved using ILOG CPLEX 12 solver incorporated in GAMS 24.1.3 software.81 A Pentium R Dual Core, with 2.5 GHz and 3GB RAM, was used to run the model. The solution was reached in 543 CPU seconds with 0% integrality gap. The model consisted of 11,360 constraints, 9,710 continuous variables, and 1,494 discrete variables. The optimal generalized supply chain network configuration is shown in Figure 4 and consists of two contracted supplier nodes (BG, RO), three P/W nodes with both capabilities established (ES, IT, BE), one P/W node with only warehousing capability established (CH), and of course 18 market nodes. The optimal network configuration is reached with a total cost of 742,016 rmu. This total cost consists of transportation cost (224,741 rmu), infrastructure cost (218,905 rmu), handling cost (130,402 rmu), purchasing cost (98,561 rmu), and production cost (69,407 rmu). It is evident that the lion’s share of cost contribution is due to transportation cost and infrastructure cost while the other cost elements are responsible for the remaining one-third. Tables 10−12 present the optimal total flows in the network. Both contracted suppliers provide materials to all established P/W nodes. All P/W nodes with both capabilities established (ES, IT, BE) allocate capacity to the one P/W node which has only warehousing capability established (CH), whereas a capacity allocation is also realized between a pair of P/W nodes with both capabilities established (ES node sends material to BE node). The P/W located in BE serves 11 markets, the one located in CH serves four, the one located in IT serves two, and the one located in ES serves only the local market. A sensitivity analysis, aiming to investigate whether and how slight changes in some input parameters affect the optimal solution, is considered valuable for our analysis. Eight analyses were conducted based on an equal number of important input max max max parameters (ainbound , aoutbound , Din, Emax in in en , Sin , Wm , Qinn , λek). Each parameter is deviated from its initial value over the range of ±10%, by different increments, and then for each one of these variations the model was solved to optimality. A total number of 48 runs were performed, and the results are summarized in Table S.7 provided in the Supporting Information material. In terms of network structure the model is robust to individual variations, ceteris paribus. In most of the cases, the optimal configuration is kept unchanged, while in cases where a change is realized this concerned the non-establishment of the P/W node, with only warehousing capability installed, and its transportation links with the other P/W nodes. In terms of optimal total cost the model reacts fairly enough in demand changes and is insensitive to all other parameters. The response of each cost element is based on the nature of the deviated input parameter. The results indicated that demand volatility seems to have a big impact on the objective function and smaller on purchasing, infrastructure, and transportation costs, respectively, while deviations in the maximum warehouse capacity parameter and the coefficient relating warehousing

5,805

TR PL DE AT NO CH BE PT FI Qinn′

Table 12. Optimal Total Flow of Material (Qinn′) from P/W to Each Market (tons/year)

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Figure 5. Optimal fixed echelon supply chain network of the case study.

Figure 6. Optimal total cost elements for GSCN and FSCN.

capacity of material i handled that leaves the P/W node have a slight effect in handling and transportation costs. Finally production cost is changed only by the deviations of the parameter related to the amount of manufacturing resource e required to perform the unit amount of operation task k. 4.2. Comparison with a Fixed Echelon Counterpart Model. In an effort to further highlight the value and strengthen the prospect of the proposed model, a comparison with a counterpart model is made that utilizes the fixed echelon network structure, shown in Figure 1, as it is the mainstream structure in the literature. In this model the hybrid P/W nodes are treated as separated entities included into two consecutive echelons, namely, plants echelon and warehouses echelon. Material flow should pass between these two echelons, and also intralayer flows are not allowed. All other constraints included

in our modeling framework, along with the objective function, remained unchanged. A representative fixed echelon model can be found in the work of Georgiadis et al.57 We must point out that the optimal network structure of the fixed echelon counterpart model is obtained freely by the optimization procedure without any fixing of binary variables expressing network nodes and/or transportation connections among them. The fixed echelon supply chain network (FSCN) design model was solved again, with the same data presented in Tables 2−9 in this work and Tables S1−S6 in Supporting Information material, and the optimal configuration is illustrated in Figure 5. The same two suppliers are contracted (BG, RO) with three plants (PT, BE, CH) and five warehouses (IT, DK, PT, BE, CH). In general, this optimal configuration is rich and 13134



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analysis results. This material is available free of charge via the Internet at http://pubs.acs.org.

complicated as it selects more nodes and establishes more transportation connections. This fact resulted in the optimal total cost where it reached 1,051,214 rmu, a figure approximately 42% higher than the optimal total cost of the generalized supply chain network (GSCN) design model. As shown in Figure 6, this cost gap is attributed mainly to infrastructure and handling cost as the FSCN model is forced by its a priori structure to build more facilities (sum of plants and warehouses) in order to satisfy customer demand and has no capacity allocation options. The FSCN model ends up in an optimal structure where the more sizable material flow connections are among the plants and warehouses that are located and build at the same country-area. More specific, the PT plant provides Portugal’s warehouse with 34,522 tons/year of material i for storage, that is 56 times bigger than the amounts provided in the other warehouse by the same plant, while BE’s and CH’s plants provide five and two times bigger amounts in Belgium’s and Switzerland’s warehouses, respectively. Moreover, the capacity of its established warehouses is higher, and this contributes to higher handling cost.

■

Corresponding Author

*E-mail: [email protected]. Notes

The authors declare no competing financial interest.

■

ACKNOWLEDGMENTS This research has been cofinanced by the European Union (European Social FundESF) and Greek national funds through the Operational Program “Education and Lifelong Learning” of the National Strategic Reference Framework (NSRF)Research Funding Program: Thales. Investing in knowledge society through the European Social Fund.

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NOMENCLATURE

Indices

e = manufacturing resources (equipment, utilities, and so on) i = materials (raw materials, intermediate products, and endproducts) k = tasks (mixing, cutting, and so on) n = nodes (suppliers, P/W nodes, and markets)

5. CONCLUSION This work introduces a modeling framework that provides flexibility options on designing and operating SCNs. The novel features of the framework are the combined role of manufacturing and warehousing capability in the same node and their inner connection along the level. The model is capable of deciding the appropriate suppliers and material flow connections including intralayer flows but mainly the location and role/capability of the generalized nodes. A generalized node can be a manufacturing plant, a warehouse, or both. Due to this fact, the model adds flexibility to the structure of the supply chain network and cost benefits to the company. The proposed MILP SCND model was evaluated in a real case study from a Europe based consumer goods company in order to illustrate its applicability. A sensitivity analysis documented the model’s robustness to slight variations on structural and operational parameters whereas a comparison with a FSCN design model revealed its superiority. At this point it should be emphasized that the core scope of the research effort is to introduce an innovative generic modeling framework for the SCND problem and implement it in a real life case study so as future works in the field might espouse it. Guided by this objective, an attempt has been made to keep the model as simple as possible and include only the core operations taking place within the SCND phase. Demand was deterministic but a simple and straightforward approach, such as a scenario tree approach, or even advanced techniques, such as the multistage stochastic programming, could be easily integrated in the modeling framework. Likewise, other important aspects, such as, inventory policy, vehicle routing, environmental impact, financial engineering, quality management, marketing strategy, reverse logistics, and special types of material such as degradable and perishable goods, just to name a few, are all worthwhile aspects that can be incorporated in the proposed generic modeling framework.

■

AUTHOR INFORMATION

Sets

C = set of market nodes Kn = set of production tasks that can be performed at P/W node n ∈ PW N = set of nodes of the network to be designed PW = set of production/warehousing nodes S = set of supplier nodes Parameters

CTinn′ = unit transportation cost of material i from node n ∈ N to node n′ ∈ N CPin = unit production cost for material i in case of established production capability at P/W node n ∈ PW CSin = unit purchase price of material i from supplier node n ∈ S CWH in = unit handling cost for material i in case of established warehouse capability at P/W node n ∈ PW CPn = annualized fixed cost of establishing production capability at P/W node n ∈ PW CSn = annualized fixed cost of establishing relationship with supplier node n ∈ S CW n = annualized fixed cost of establishing warehouse capability at P/W node n ∈ PW Din = demand for material i by the market node n ∈ C Emax en = maximum rate of availability of resource e at P/W node n ∈ PW Emin en = minimum rate of availability of resource e at P/W node n ∈ PW Qmax inn′ = maximum rate of flow of material i that can be transferred from node n to node n′ Qmin inn′ = minimum total rate of flow of material that can be practically and economically transferred from node n ∈ N to node n′ ∈ N Smax = maximum rate of availability of material i by the in supplier node n ∈ S Smin = minimum rate of availability of material i by the in supplier node n ∈ S Wmax n = maximum warehouse capacity that can be established at P/W node n ∈ PW

ASSOCIATED CONTENT

S Supporting Information *

Tables S1−S7 listing fixed cost for establishing warehouse capability, unit handling cost of each material, demand for each material, unit transportation cost of material, and sensitivity 13135



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Wmin n = minimum warehouse capacity that can be established at P/W node n ∈ PW ainbound = coefficient relating warehousing capacity at P/W in node n ∈ PW to flow of material i handled that enters the node aoutbound = coefficient relating warehousing capacity at P/W in node n ∈ PW to flow of material i handled that leaves the node γPen = cost to install a unit of manufacturing resource e at P/ W node n ∈ PW γW n = cost to install a unit amount of warehousing capacity at P/W node n ∈ PW δPen = unit cost of consumption of manufacturing resource e at P/W node n ∈ PW λek = amount of manufacturing resourcee required to perform unit amount of operation task k νik = amount of material i produced by unit amount of operation task k

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Continuous Variables

Een = total rate of availability of manufacturing resource e at P/W node n ∈ PW Qinn′ = rate of flow of material i transferred from node n ∈ N to node n′ ∈ N Sin = rate of availability of material i by the supplier node n ∈ S Wn = warehouse capacity established at P/W node n ∈ PW ξkn = rate of operation of task k at P/W node n ∈ PW Binary Variables

Yn = 1, if a P/W node n ∈ PW is to be established; 0, otherwise Ypn = 1, if a production capability is to be established at P/W node n ∈ PW; 0, otherwise YSn = 1, if a supplier node n ∈ S is selected to provide material to the network; 0, otherwise YW n = 1, if a warehouse capability is to be established at P/W node n ∈ PW; 0, otherwise Xnn′ = 1, if material flow is to be transported from node n ∈ N to node n′ ∈ N; 0, otherwise

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Optimal Design of Multiechelon Supply Chain Networks with

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