Supply Chain Management with Optimal Scheduling - Industrial

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Supply Chain Management with Optimal Scheduling A. C. S. Amaro† and A. P. F. D. Barbosa-Po´ voa*,‡ Inst. Sup. Cont. Administrac¸ a˜ o, ISCAC, Quinta Agrı´cola, 3040 Coimbra, Portugal, and Centro de Estudos de Gesta˜ o do IST (CEG-IST), Instituto Superior Te´ cnico, UniVersidade Te´ cnica de Lisboa, AV RoVisco Pais, 1049-001 Lisboa Codex, Portugal

In this paper, a new formulation is presented for the optimal scheduling of industrial supply chains. This is accomplished through the development of a master representation model, the ChainSTN, that unambiguously describes the supply chain topology, resources, operations, and involved materials. Also, different market opportunities (supply, demand, and price options) and manufacturing and transport considerations are regarded. By combining these concepts with a discrete and uniform representation of the scheduling time domain, a mixed-integer linear programming formulation (MILP) is attained. A detailed optimal scheduling plan is obtained, where different (i) operational decisions, describing the operations of the entire partners network; (ii) transportation policies, detailing the material flows among partners; and (iii) market options (provider’s conditions and customers’ demand orders) are jointly scheduled so as to optimize a global economical criterion, in here, the supply chain global profit. The flexibility and applicability of the new formulation are validated through the solution of an industrial example where different operational scenarios are discussed. 1. Introduction How to go further in today’s highly competitive market is a major common concern in any international or national organization. The observed enlargement of markets encourages companies to look into new, while geographically disperse, business opportunities. These require broad business and operational strategies not captured by traditional managing perspectives. In fact, for many years, companies managed their logistics process, procurement, production, and distribution in a nonintegrated way.1 However, the current challenges stimulate functional integration, and organizations do now consider wide management attitudes to enhance global business and operational integration. These include, namely, multienterprise partnership and supply chains (SCs). In a recent paper, Grossmann2 provides an overview regarding the process industry developments and identifies some major challenges to the new millennium. Three major research challenges are identified: product discovery and design, enterprise and supply chain optimization, and global life cycle assessment. This paper looks in particular to the area of supply chain optimization. This area is attracting a great interest from industry and academia. From an industrial standpoint, competitiveness, service levels, and economical performances are some of the major drivers.3 The aim is placed on flexible SC configurations4,5 and optimized operational plans6-8 to reach a close commitment to cost and products quality, life cycle, variety, and price. Concerning the academic background, the optimization of SC involves a variety of decision problems with different time horizons and goals. These span from the strategic to the operational level, passing through the tactical level. Most of the research work addresses supply chain management (SCM) problems from a strategic or a tactical point of view. The major challenges include the development of models for strategic9 and tactical planning.10-12 Their industrial implementation results frequently in large-scale optimization problems, and thus, some * To whom correspondence should be addressed. Tel.: +351214233265. Fax: +351214 233 568. E-mail: [email protected]. † Inst. Sup. Cont. Administrac ¸ a˜o, ISCAC. ‡ IST, Universidade Te ´ cnica de Lisboa.

modeling difficulties and computational burdens may turn them unsolvable within reasonable optimality gaps. The dimensional scale of these problems is, to some extent, explained by the complex SC configuration and associated operability characteristics. The SC multifunctional nature is ensured by a complex network of operational stages within geographically disperse entities (SC members) and involves a large number of operations, resources, and materials as well as complex partnership relations. Therefore, industrial management strategies and research contributions agree on the relevance of coordination of decisions within the SC members at different operational stages, from suppliers till distribution to end customers. Nevertheless, the majority of the foregoing academic work considers SC stages apart from each other or at least addresses simply the integration of only some of the SC stages13 as follows: buyer-vendor, inventory-distribution, or productiondistribution.10,11,9 Furthermore, despite the improvements on information systems,14 several coordination issues are still ignored and SC management is often supported by nonoptimized logistic approaches (with large material buffers) to entail the integration of SC stages. As Maloni and Benton15 report, despite the extensive conceptual background in SCM, very few researchers attempt rigorous analytical approaches. The existent sparse contributions do not support the entire chain and are limited to the coordination of just some of the many SC functions. The need for future research work on any of the SC stages is recognized by different research perspectives, ranging from logistics, manufacturing, and business to the financial area.16 Some lack remains on deterministic modeling advances, solution methods, and decision-support tools. Furthermore, concerning the organizational perspective, in an SC, two managing approaches may be considered: centralized and decentralized. Centralized approaches consider the existence of a central entity owning a global visibility on demand, costs, and inventory status at all system locations. This “core entity” tries to smooth out the pressure among SC members, and an attempt is made to optimize the entire chain. Instead, decentralized approaches entail decisions made independently by separated members.17

10.1021/ie070262a CCC: $40.75 © 2008 American Chemical Society Published on Web 11/30/2007

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Figure 1. Supply chain topology.

Also, Fleischmann et al.18 noticed that a large amount of the research efforts have been placed on discrete product industries while only a few contributions within Supply Chain Optimization (SCO) deal with process industries. This is also accomplished by the empirical study of French and LaForge,19 covering a large survey of U.K. process industries, such as chemicals, food, plastic, and rubber, that emphasize the need for future research on process industry networks’ at the design, inventory, planning, and scheduling. Recently, Shah20 presented the state of the art in SC design and operation for the process industries, and again, emphasis was on the integration relevance on aspects such as business and physical processes, topology and operations complexity at different time scales, strategic decisions and uncertainties, sustainability, and environment, among others. To enable some of these integrative challenges, several hybrid modeling approaches (Mestan et al.21) and simulation methodologies (Mele et al.17 and Reiner22), inspired on system control concepts and on object-oriented architectures (Hung et al.23), have recently been proposed. Another important remark is that most of the developments in SCO almost disregards transportation operations, even when the SC operational coordination is the major goal. As Lee and Chen24 discussed, most machine scheduling models consider either that there are an infinite number of transporters for delivering jobs or instead jobs are delivered instantaneously from one location to another. No transportation time nor capacity considerations are addressed.25,26 In conclusion, a high number of formulations covering a large spectrum of SC aspects have been proposed so far. However, some space remains when trying to globally integrate SC multifunctional and geographically disperse structures while attempting to determing accurate modeling approaches and operational details. In this paper, the SC operation is considered in an integrated form and a new formulation is developed for the detailed optimal scheduling of SC. This has the advantage of combining the SC topology, the processes, the operability requirements, and the different transportation policies and market conditions into a single framework. Within the topology, an enlarged set of SC sites (partners) with different geographical locations is considered as well as the available connectivity network. In terms of

operability, different operational policies at the production, storage, distribution, and transportation are analyzed. Also, the materials requirements imposed by market conditions (providers’ settings and customers’ demand orders) or predefined processing considerations are explicitly integrated. The problem is formulated as a mixed-integer linear programming (MILP) model that is solved using a standard branch-and-bound procedure. The rest of the paper is structured as follows. First, the supply chain structure is characterized and its operability is described. Then, a new representation model, the ChainSTN, is presented followed by a detailed description of the associated mathematical formulation. The solution of an industrial example is explored afterward where the applicability and flexibility of the proposed model are shown. Finally, in the last section, some conclusions are drawn and some future improvements are discussed. 2. Supply Chain Operation and Representation Details A supply chain is as a master operational network with a complex multifunctional configuration, involving geographically disperse resources, materials, and markets. A multistage description is frequently adopted to represent the SC operational network. However, as was formerly mentioned, most modeling approaches do not support the entire chain description and consider just some of the multiple SC functions. Thus, some economic and operational-scale savings are lost or at least less explored than the expected. In this paper, the SC multistage description is generalized to explicitly account for the integration of the entire partners’ network. This is attained through the development of a new representation model. Concerning the SC operation, a former concept emerges from the link between the geographical location of SC partners and the available resources’ suitability. Supply chain resources are classified into general processing resources (e.g., transformation equipments as reactors, fillers, etc.; or for inventory purposes, tanks, vessels, warehouses, etc.) and transportation resources (e.g., vehicles, trucks, etc.). Processing resources are installed at specific geographical locations and define a site within the SC structure (e.g., warehouse S1, plant I1, etc.; see Figure 1). Each site is responsible for the fulfillment of a given purpose within the master operational plan (e.g., store material A,

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Figure 2. Description of material flows.

produce material C, etc.). The SC sites are grouped into clusters based on their operational objectives, suitability, and functional similarity Thus, a set of clusters is generated to represent the SC operation. Four major clusters are defined, namely, (i) supply (S1-S5), (ii) production (I1 and I2), (iii) customization (packing and labeling, PL1 and PL2), and (iv) distribution (DC1-DC10). These are connected by a master connectivity structure. The SC operation is performed all along these sets of clusters with transportation resources ensuring the material flows among cluster sites (partners) and into/from the external markets (Figure 1). The diagram in Figure 2 describes generically the material flows involved in the SC operation. The SC operation results then in a complex macrotask strongly dependent on the coordination issues, partnership relations, operational configuration, processes, transportation, resources, and materials, as well as on the market options, customers and providers, due dates, and capacities. Moreover, the connectivity network is represented by a set of transportation structures. These are groups of autonomous transportation resources, (e.g., vehicles, trucks, train wagons, boat containers, etc.) characterized by the following: (i) a capacity, defined by the set of enclosed transport resources; (ii) a common materials’ suitability; (iii) an ownership criteria, inhouse or contracted resources; and (iv) an operating mode, dedicated or shared policy. Transportation structures can be owned by a chain partner (inner capacity), or alternatively, they can be a service provided by a third-party logistics company (contracted capacity). Thus, different transportation decisions are allowed. SC partners can decide on using exclusively their own transportation capacity or, instead, they may contract the whole or a fraction of the required transport capacity, using it in a dedicated mode or sharing it with other partners. On the basis of the discussed SC details and characteristics, a new representation approach, the ChainSTN, is proposed; see Figure 3. The equipment State Task Network, eSTN (BarbosaPo´voa28) and the Flowpaths (Amaro and Barbosa-Po´voa27) modeling concepts, proposed for multipurpose batch plants, are adapted and generalized to the SC context. Three main components are, therefore, accounted at the ChainSTN repre-

Figure 3. Details of the proposed ChainSTN representation.

sentation: (i) the structure, (ii) the materials, and (iii) the operations (Figure 3). The structure represents the chain topology and considers the chain sites, the market places (providers and customers), and the associated connectivity network. As was formerly reported, two major sets of resources are considered within the chain topology: process resources (j ∈ J) and transportation resources (V ∈ V). The former are available at the supply chain sites and ensure the execution of processing operations (tasks), while the second ones are responsible for the transportation of materials (transportation flows) among SC partners and are grouped into transportation structures (π ∈ Π). Also, a set of materials states is considered to represent all the available materials and their SC location. Each material state defines the link between a material and its specific location (e.g., sA1 is raw material A at Gm1 location, sA2 is raw material A at Gm2 location (Figure 4a)). Finally, the operations describe all feasible events (transformation, storage, transport, etc.) performed to guarantee the SC operational goal. The occurrence of any operation requires the allocation of a suitable SC resource, the availability of the material states defined within the operation recipe, and the fulfillment of any operational preconditions. Thus, the ChainSTN approach considers then the SC operation as a globally coordinated superstructure described by three major network structures: the process, the connectiVity, and the market networks. The process network involves a set of processing eVents that account for both the transformation (e.g., reaction i, performed at reactor j: j ∈ JP ⊆ J) and the storage events (e.g., material state s stored at tank j: j ∈ JS ⊆ J). Each processing event is denoted as a task i that defines the linkage between a processing operation and a suitable resource j to execute it. The following properties are assumed for each task occurrence: (a) single assignmentseach processing resource (j ∈ JP) is assigned to only one task (i ∈ I) at any time t (e.g., reactor j is allowed to perform a single reaction at time t); (b) task recipesany task i, performed at resource j, consumes a certain amount of input materials, s ∈ Sin i , and produces another amount of output materials, s ∈ Sout i , both defined by the task recipe, after pti fixed time units (see work of Barbosa-Po´voa28 for further details). On the other hand, the transportation flows (or simple flows) represent the transport operations. Each flow l is defined by the following: (i) a material to transport, (ii) a transportation structure (set of transport resources), (iii) a chain path, and (iv) a transportation time (e.g., flow l1 represents the transport of

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Figure 4. (a) Transportation flows’ background and (b) flows’ details.

raw material A, by transportation structure π, between warehouse S1 and plant I1 and spent 2 h; see Figure 4b and paper by Amaro and Barbosa-Po´voa27). A set L is generated to store all feasible transportation operations (|L| ) NL flows) that represent the materials mobility within the SC network. The occurrence of any of these flows depends on the availability of a suitable transport resource, V, within the transportation structure π to which it belongs, V ∈ Γπ ⊆ V. As opposed to the occurrence of processing tasks, the sharing of transportation resources by more than a flow is considered. This sharing of resources is, in practice, constrained by the materials to be merged within a common piece of equipment as well as by the chain path considered. Thus, an incompatibility criterion is considered to account for the simultaneous assignment of transport operations. This is defined as follows: any two transportation flows are incompatible if (a) the associated materials are incompatible (e.g., physical incompatibility, solid A and liquid B; incompatible customizations, bulk and packed materials, etc.); or if (b) they flow in different directions, even with compatible materials (e.g., a flow l1 transferring material A from warehouse S1 to I1 plant is incompatible with flow l2, responsible for the transport of material A from S1 to I2 plant; see Figure 4b). An incompatibility matrix, Θ ) [θll′]l,l′)1,...,NL, is then generated to store the information on the incompatibility condition between any pair of transportation flows (l, l′). Matrix Θ is defined as follows:

operations (SC eVents) within a global and integrated approach. Three interrelated networks, the process, the connectiVity, and the market networks (Figure 3), are combined at a single SC master representation. Also, different operational decisions and precondition requirements are considered within each one of these networks.

Θ ) [θll′]l,l′)1,...,NL: θll′ ) 1 if transportation flows l and l′ are incompatible ∀l,l′ ∈ L 0 otherwise

• The whole set of involved materials and their availability, requirements, and geographical locations;

{

The resources available within a given transportation structure π (V ∈ Γπ ⊆ V) have the same incompatibility characteristics. Therefore, the simultaneous operation of more than a transportation flow (l, l′) sharing a given transportation resource requires flows compatibility (θll′ ) 0, l, l′ ∈ L: l * l′). Incompatible flows (l, l′ ∈ L: θll′ ) 1) can only occur at a time t, if they are allocated to different resources. Finally, the market network describes all feasible material exchanges arising between the SC entities and the outside market. These account for external receipts (e.g., supply of materials or services from external providers into SC sites) or deliveries (e.g., material exchanges from SC sites into market customers). Different market options such as contracted and noncontracted receipts or deliveries, due dates, and capacity requirements are accounted. In summary, the ChainSTN is a master network representation that combines (1) the structure, (2) the materials, and (3) the

3. Problem Definition On the basis of the SC characteristics formerly described, the scheduling problem is stated as follows. Given: • The scheduling horizon and all the time conditions involved (e.g., working timetables, programmed time schedules for receipts, deliveries’ due dates, etc.); • The Chain topology characteristics as follows: - Locations: geographical map of all the SC positions (resources and markets); - Resources: available processing (transformation and storage) and transportation equipment and facilities and their capacities, suitabilities, and operational preconditions, if any; - Markets: set of external suppliers (material or services providers) and customers (demand clusters or positions), their capacities, and materials suitability;

• The description of the operational networks as follows: - Process network: set of SC processing operations (transformation and storage tasks), their resources suitability, the associated recipes (defining the involved material states), and the processing time and any existent operability conditions; - ConnectiVity network: set of SC transport operations (transportation flows) described in terms of the material to be transported, the suitable transportation structure, the chain path, and the travelling or transportation time and any existing preconditions; • The details of the market network as follows: - ProViders: characterization of the existing supply conditions, namely, the materials, the associated time tables, and the corresponding amounts; - Customers: delivering options to each material, the corresponding due dates and amounts, and the fulfillment requirements, if any.

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Determine: • The global schedule of the SC network while accounting for a detailed operational plan at the processing, storage, transportation, and market exchanges, during all of the scheduling horizon, So as to: • Maximize the supply chain net global profit. This considers a detailed balance of costs and incomes that account for different requirements on the resources, materials, and operations as well as on different market options and opportunities during the whole scheduling horizon. 4. Problem Formulation The supply chain operation is characterized, as previously discussed in Section 2, by a complex operational structure with a geographically disperse network of resource materials and market places. On the basis of the proposed representation, ChainSTN, a mathematical formulation is developed. The following definitions and assumptions are considered: • An uniform time grid is used to represent the time domain. This corresponds to the split of the scheduling horizon, H, into a discrete number of elementary time steps of fixed length, ∆t, from t ) 1 to H + 1. • All supply chain events (processing tasks, i, and transportation flows, l) have a fixed completion time (pti and ttl, respectively) defined as an integer multiple of the elementary time step ∆t. • The assignment of any event, i or l, to a suitable equipment resource, j or V, is allowed at the intervals’ boundaries (time grid points) and not between them. • For tasks and transportation flows, a non pre-emptive mode of operation is considered. Therefore, any of these events once started cannot be interrupted. • Two transportation flows, l, l′ ∈ L, defined over different transportation structures, π, π′ ∈ Π, are compatible, unless another operational specification is given. • No operation assignment can be performed before the beginning of the scheduling horizon (t < 1) nor after its finish (t > H + 1). A mixed-integer linear programming (MILP) formulation is developed. This comprises the following: (1) a set of linear constraints, describing the SC topology and the networks of materials, operations (processing and transportation), and markets; and (2) a linear objectiVe function, responsible for the evaluation of an economical criterion, assumed as the global SC net profit. The formulation constraints and the objective function are defined based on a set of: • binary variables, YλVlt and Ypijt, accounting for the assignment of events (l ∈ L and i ∈ I) to the suitable resources (V ∈ V and j ∈ J), at any time t, and • continuous or integer Variables to represent the following: (i) the capacity of the resources allocated to each suitable event, at any time t (QλVlt, Qpijt, and Sst) and (ii) the market receipts/deliVeries defined for each material state, s, at any time t (Rst and Dst). The integrality conditions on these variables are based on the resources’ operability characteristics (e.g., bulk or packed productions) and on the associated nominal units (e.g., mass, volume, or discrete material units, etc.). The model constraints are divided into five major groups: (i) operations network (operability and capacity constraints), (ii) market settings and options (deliveries and receipts), (iii)

time-scheduling requirements, (iv) material balances, and (v) scheduling-horizon boundaries. 4.1. Formulation Constraints. 4.1.1. Operations Network. This group of constraints accounts for the conditions that should be observed during the assignment of the SC operations. It involves two main sets of constraints: (a) operability and (b) capacity constraints. 4.1.1.a. Operability. These constraints include all the operational requirements arising from the assignment of SC events (tasks and flows), independently of any resource capacity consideration. The following relations are accounted for: (1) incompatibility, (2) continuity, and (3) resource sharing. 4.1.1.a.1. Incompatibility. A set of constraints is defined for transportation flows, l, and processing tasks, i, so as to guarantee that incompatible assignments, representing infeasible operations, do not occur. Therefore, a nonoverlapped operation must be ensured for the occurrence of incompatible flows l and l′, within a common transportation resource, V, and for the allocation of processing tasks i to the same processing resource j (i ∈ JIj). For transportation flows, this operational requirement is stated as follows: t-ttl+1

∑

t-ttl′+1

∑

YλVlt′ +

t′)t

YλVl′t′ e 1

∀ π ∈ Π, ∀ V ∈ Γπ

(1)

t′)t

∀ l,l′ ∈ Fπ: l′ > l ∧ θll′ ) 1, ∀t ) 1, ..., H As has been discussed in Section 2, the incompatibility matrix, Θ, is symmetric (Θ ) ΘT), and therefore, only the elements above the diagonal (l, l′ ∈ Fπ ⊆ L: l′ > l, ∀π ∈ Π) are considered. On the other hand, for the processing operations and at a given time t, the assignment is defined through the occupation of a suitable processing resource j. This can be either idle or processing a single tasks i. t-pti+1

∑ ∑ Yp t′)t

ijt′

e1

∀j ∈ JP, ∀t ) 1, ..., H

(2)

i∈JIj

4.1.1.a.2. Operations Continuity. This set of mathematical relations is developed exclusively for the fully compatible flow events (l, l′ ∈ L: θll′ ) 0) and guarantees that, if a flow occurs, it cannot be interrupted along its time of transport (nonpreemptive mode of operation). t-ttl+1

∑ t′)t

YλVlt′ e 1

∀π ∈ Π, ∀V ∈ Γπ, ∀l ∈ Fπ: ¥πl ∪{l} ) Fπ, ∀t ) 1, ..., H (3)

For the remaining events, processing tasks i, or transportation flows l, having at least one incompatible flow, l′, defined over the same transportation structure π (flows to whom proposition ∃ l, l′ ∈ Fπ ⊆ L: θll′ ) 1 is true), the incompatibility constraints previously defined (eqs 1 and 2) ensure both the operational compatibility and the continuity conditions. 4.1.1.a.3. Resources Sharing. As was formerly stated, only transportation resources are allowed to perform several, while suitable, operations within a common time frame. These inequality constraints are responsible for the feasibility requirements involved with the resource-sharing occurrences. Therefore, any two flows, l and l′, defined over the same transportation structure, π ∈ Π, eVen compatible, can only share

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a transport resource V if they are assigned within the same time frame: 1

∑ Qλ

t-ttl′+1

∑ ∑

ϑlπ l′∈¥πl

(iii) Feasibility boundssminimal capacity bounds to any flow assignment:

YλVl′t′ + YλVlt e 1

Vl′t

+ QλVlt g CVVΨVYλVlt

l′∈¥πl

∀π, ∀V ∈ Γπ, ∀l ∈ Fπ, ∀t ) 1, ..., H (9)

t′)t-1

∀π, ∀V ∈ Γπ, ∀l ∈ Fπ : ttl > ∆t, ∀t ) 1, ..., H (4)

The double summation is normalized based on parameter ϑlπ that accounts for the number of flows l′ compatible with flow l, at transportation structure π (cardinal of set ¥πl). No set of constraints is required for those flows l that spent a single time unit (ttl ) ∆t) since this is accounted intrinsically into the formulation. 4.1.1.b. Capacity Constraints. These mathematical relations ensure the feasibility between the assignment of events and the capacity bounds imposed by the associated equipment resources. 4.1.1.b.1. Task Events. The assignment of any processing resource, j ∈ JP, to a suitable task, i ∈ JIj, has to be performed within a feasible resource occupation. This is defined by a minimal equipment occupation, CJjΦmin ij lower capacity bound, upper capacity and by a maximal resource fulfillment, CJjΦmax ij bound. These depend on the resource equipment and, for each resource j, on the assigned operation, i. (i) Batch operations characterized by nominal capacity units (e.g., mass, volume, etc.): min CJjΦi,j Ypijt e Qpijt e CJjΦmax ij Ypijt ∀j ∈ JB, ∀i ∈ JIj, ∀t ) 1, ..., H (5)

(ii) Continuous or semicontinuous operations measured by processing rates (e.g., mass, volumetric, or discrete rate): max CJjptiΦmin ij Ypijt e Qpijt e CJjptiΦij Ypijt ∀j ∈ JP\JB, ∀i ∈ JIj, ∀t ) 1, ..., H (6)

Hence, if a rate is measured, the resource capacity, CJj, is multiplied by the task processing time, pti. Note that these processing times can be either a single time interval or an integer multiple of that interval, as was formerly considered for batch tasks. If no prior requirements are defined to the operation of these resources nor for the handled materials, the processing times are settled equal to a time unit (pti ) ∆t). 4.1.1.b.2. Flow Events. The transportation of materials can be performed using (i) a dedicated transport decision (a single flow) or (ii) a transportation sharing option (assignment of multiple flows). In both cases, the allocation of the transportation flows has to be done within feasible capacity limits. A full equipment charge is considered as upper capacity limit and a minimal equipment occupation is defined for the lower capacity bound. These capacity requirements are stated as follows: (i) Dedicated transport decisionssingle flow assignment: QλVlt e CVVYλVlt

∀π, ∀V ∈ Γπ, ∀l ∈ Fπ, ∀t ) 1, ..., H

(7)

(ii) Transportation share optionssimultaneous allocation of compatible flows to a single suitable piece of equipment:

∑ Qλ

Vlt

l∈Fπ

e CVV

∀π, ∀V ∈ Γπ, ∀t ) 1, ..., H

(8)

The lower capacity bounds CVVΨV are defined for both transportation decisions, dedicated and shared. The parameter ΨV represents the minimal percentage of resource’s V capacity that justifies any transportation flow assignment (minimal amounts of materials that have to be charged into a transportation equipment to allow the occurrence of a transport operation). Moreover, a set of equality constraints are considered to account for the global amount of material transported by each flow l, through the set of suitable transportation resources. QΛlt )

∑ Qλ

Vlt

∀π, ∀l ∈ Fπ, ∀t ) 1,...,H

(10)

V∈Γπ

4.1.1.b.3. Storage Events. Storage events represent inventory operations, involving a defined material state, s, and a suitable resource unit (vessels, tanks, containers, etc.) or facility (warehouse, space floor, etc.), j ∈ JS. Inventory decisions account for shared and dedicated operations on storage resources. In either, the amounts to be stored must be within max suitable resource capacity bounds, CJjφmin sj and CJjφsj . max φmin sj CJj e Sst e CJjφsj ∀j ∈ JS, ∀s ∈ KSj, ∀t ) 1, ..., H + 1 (11)

These capacity relations are not required for inventory operations represented by processing tasks, since constraint 5 ensures that the amount to be stored fulfills the storage capacity limits. When a storage resource j is shared by a set of suitable material states, s ∈ KSj: |KSj| * 1, the total amount of materials stored must be within the available resource capacity.

∑S

st

e CJj

∀j ∈ JS, ∀t ) 1, ..., H + 1

(12)

s∈KSj

4.1.2. Market Settings and Options. Different market requirements and management options, at the providers’ and customers’ positions, are considered to characterize the material exchanges among SC partners and from/into the external market. These include the following: (a) material deliVeries and receipts and (b) other serVices exchanges as the contract of transportation capacities to third-party logistic companies (external providers). 4.1.2.a. Delivery and Receipt Options. In this, the decisions on the material flows between SC partners and market providers/ customers are attained. • DeliVeries, Dst: These represent the transfer of materials from a SC site into a customer’s position. Delivery operations are bounded by the settings of customers’ demand Up orders, (DLow ˆ Low ˆ Up st , Dst ) and (D st , D st ), and they also depend on the transportation options accomplished. Thus, deliveries can be: - performed by the customers. In this case, no transportation requirements have to be considered and only demand bounds need to be observed. DLow e Dst e DUp st st

∀s ∈ SD, ∀t ) 1, ..., H + 1

(13a)

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- done by the SC partners using the aVailable transportation resources. Here, some additional transportation flows have to be considered, l ∈ SLout s , and thus, transport capacities should be accounted. e D ˆ Low st

∑

QΛlt e D ˆ Up st

∀s ∈ SD, ∀t ) 1, ..., H + 1

l ∈SLout s

YνVt -

(14a)

The upper and lower bounds of these inequality constraints are defined independently for each delivery option. Therefore, a combined strategy involving both delivery options can be obtained since these are not mutually exclusive operations. Instead, if customers should decide on a single delivery option at any time t of the scheduling horizon, these events should be considered to be mutually exclusive. This can be accomplished explicitly at the model formulation through the integration of constraints, YDst DLow e Dst e YDst DUp st st ˆ Low e (1 - YDst )D st

4.1.2.b. Other Services’ ExchangessContract of Transportation Capacities. Independently of any cost consideration,contracted capacities are evaluated based on the transportation resources’ utilization.

∑

∀s ∈ SD, ∀t ) 1, ..., H + 1

(13b)

QΛlt e (1 - YDst )D ˆ Up st

l∈SLout s

∀s ∈ SD, ∀t ) 1, ..., H + 1 (14b) where an additional set of binary variables, YDst , is required to define the delivery option of each material state s at each time interval t. If YDst ) 1, delivery operations are performed by customers without using any transportation requirements of the SC partners. Otherwise, the material deliveries should be performed exclusively by the SC partners, since no materials’ delivery is accounted for by customers, YDst ) 0. These delivery scenarios can also be integrated as predefined options (problem Low )0∧D ˆ Up )0∧ data) by considering (D ˆ Low st st ) 0) ∨ (Dst Up Dst ) 0). Moreover, constraints 13a, 13b, 14a, and 14b consider a just-in-time, JIT, transportation policy. However, other considerations can be accounted for through the decision variables representing the material deliveries, Dst and QΛlt. These can be defined, namely, by the integration of event points accounting for the transportation times, Dst-ttl and QΛlt-ttl, respectively. • Receipts, Rst: These describe the materials’ exchanges coming up from the external providers into SC positions. The materials’ supply is, at any scheduling time t, bounded by the providers’ capacities, RLow and RUp st st . Different decisions on the receipts of materials are considered: - Scheduled supplies: minimal and maximal capacities are defined for each material state’s provider, s, and time interval t, within the scheduled period or predefined time frame, TRs : RLow e Rst e RUp st st

∀s ∈ SR, ∀t ∈ TRs

(15)

- Nonprogrammed supplies: they represent materials’ supplies other than the scheduled ones. A discrete number of materials’ charges, NChs,t′, with a fixed dimension, QChs is attained: NChst′ g

Rst′ QChs

∀s ∈ SR, ∀t′ ∉ TRs

(16)

1

t-ttl+1

∑ ∑

Ξπ l∈Fπ

t′)t

YλVlt g 0 ∀π ∈Π\Πowner, ∀V ∈ Γπ, ∀t ) 1, ..., H (17)

The assignment of any flow l, defined over a contracted transportation structure (π ∈ Π\Πowner), at any time t, turns the binary variable YvVt into 1, and so a transportation resource utilization is considered. 4.1.3. Time-Scheduling Requirements. In an SC, partners can have different working practices involving distinct time periods. Two major working options are considered: (i) discrete (e.g., 8 h daily) and (ii) continuous (24 h daily) timetables. In the former, the scheduling horizon corresponds to a discrete replication of a common time frame or scheduling period, T, 2T, ..., H. Therefore, some feasibility constraints, on tasks and transportation flows, are considered so as to account for a nonpre-emptive mode of operation. t-ttl+1

∑

∀π, ∀V ∈ Γπ, ∀l ∈ Fπ: ttl > ∆t, ∀t )

Yλijt′ e 0

( )( )

t′)t-1

T ∆t

, ..., NH

T

∆t

,

T

∆t

(18)

t-ptl+1

∑

∀j ∈ JP, ∀i ∈ JIj: pti > ∆t, ∀t )

Ypijt′ e 0

t′)t-1

T ∆t

( )( )

, ..., NH

T

∆t

,

T

∆t

(19)

These inequality constraints ensure that no equipment resource, V or j, is assigned to a feasible event l or i with a duration greater than the discrete time unit (ttl > ∆t or pti > ∆t) at any time t′, before a period boundary, at a distance less than or equal to ttl - 1 (or pti - 1 for task i) time intervals from each of those period boundaries, t′ ∈ {t - ttl + 1, ..., t - 1}or t′ ∈ {t - pti + 1, ..., t - 1}. No additional constraints are required for the continuous schedule option, since it is formulation intrinsic. 4.1.4. Material Balance Constraints. For each material state s and at any time interval t of the scheduling horizon, a material state equation is defined. These constraints relate the amount of each material in the state, Sst, with the material existences in the prior time interval, Sst-1, and with the amounts being produced, Qpijt-pti, consumed, Qpijt, and transferred, QΛlt-ttl, QΛlt, Dst, and Rst, into/from the state at the considered time. Sst ) Sst-1 +

∑ ∑

∑

j∈JP i∈JIj∧i∈SIin s

∑ QΛ -∑ ∑

QΛlt-ttl -

l∈SLin s

Qpijt-ptiRsi

lt

+

l∈SLout s

QpijtRsi + Rst - Dst

j∈JP i∈JIj∧i∈SIout s

∀s ∈ S, ∀t ) 1, ..., H + 1 (20)

For the initial period (t ) 1), the initial inventory on the material is given by the problem data (Ss0). Also, note that, accordingly with the measuring units used for QΛ and Qp, different conversion factors may be required. The following diagram, Figure 5, is used to illustrate the evaluations accounted at the bill of materials, BOM. Let’s

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4.2.1.b. Deliveries. Here, the incomes resulting from the materials sold to customers are considered.

Figure 5. Illustrative example of materials’ balance.

consider the material balance, at time t ) 4, for events marked from 1 to 4. Events 1 and 2 are balanced at t ) 4 in terms of the produced/released products, but their materials consumptions were considered at times t ) 1 and t ) 2, respectively. On the other hand, event 3 is balanced for the consumed/charged materials at t ) 4 but not for the produced products since the event has not yet finished at t ) 4. Finally, event 4 is not considered at t ) 4 for consumption/charge nor production/ release of materials/products, since it has started at t ) 3 and will finish at t ) 5. 4.1.5. Scheduling-Horizon Boundaries. The formulation allows the integration of different initial and final conditions to the model events, tasks, transportation flows, and storage. Different amounts of each material state, s, can be defined immediately before the beginning of the scheduling horizon (t ) 0) as ∀s ∈ S′

Sst|t)0 ) SIs0

(21)

Also, some ending conditions reporting specific amounts for the final storage levels can be integrated: (Sst|t)H+1 ) SFsH+1) ∨ (Sst|t)H+1 g SFsH+1)

∀s ∈ S′ (22)

For the remaining events, tasks i and transportation flows l, no occurrence is allowed before the beginning of the time schedule, t < 1, nor after the beginning of the last time interval, t > H. So, the binary variables are placed to zero at time instances t ) 0 and t ) H + 1, Ypijt|t)0 ) Ypijt|t)H+1 ) 0, ∀i ∈ I, ∀j ∈ J, and YλlVt|t)0 ) YλlVt|t)H+1 ) 0, ∀l ∈ L, ∀V ∈ V. Note that no decision variable is created for t < 0 nor for t > H + 1. 4.2. Objective FunctionsProfit Analysis. As was formerly reported, the objective function considered is the maximization of the SC global profit. This represents the economical balance between the (1) incomes and (2) costs of the whole SC entities.

4.2.2. Costs. 4.2.2.a. Receipts. These balance the amounts due to the external receipts of materials:

As was previously discussed in Section 4.1, constraints 15 and 16, different decisions are considered for the materials’ supply into the SC sites, namely, scheduled and nonprogrammed supplies. These are balanced at different prices or economical values. Raw materials’ supplies are evaluated at the source price (pws) for scheduled orders, while for nonprogrammed supplies, an add-on cost (CChs) proportional to the number of discrete charges (NChs) is considered. Also, the remaining material states’ receipts, if feasible, are evaluated at a market retailing price, pms ) pws(1 + ∆pws). 4.2.2.b. Storage Costs. These cost issues comprise fixed and variable terms. The former (FSCj) is independent of the resource usage and accounts for fixed expenses resulting from owned capacities, as equipments’ maintenance (vessels, tanks, warehouse spaces, etc.). Variable costs depend on the amount of each material stored, Sst, its unitary storage cost, SCs, and the predefined storage period (Ts, a time interval, a day, a week, a month, etc.). Two approaches are considered for the evaluation of storage periods, Ts. Thus, if a time interVal is considered, the costs are balanced by the stored amounts defined by the material balances (eq 20); otherwise, if the storage period, Ts, is defined as a set of time interVals, the amounts of each material state stored during the period are balanced by the arithmetic mean of the amounts crossing the time intervals defined within the storage period bounds (second option in the equation below).

4.2.1. Incomes. 4.2.1.a. Material Assets. Material assets represent the material states produced but kept in storage. The economical incomes of material assets are calculated at the exwork or producing price, pws.

∑ pw (S s

s∈S′

sH+1

- Ss0)

4.2.2.c. Supply Chain Events. The economical balance of any SC event involves fixed and variable cost terms. However,

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Figure 6. Details on supply chain transportation structures: suppliers to plants; plants and suppliers (S4 and S5) to PL sites and PL to DC sites.

cost parameters are event-dependent issues; thus, the following results: - In-house transportation capacities: in this case, fixed and variable costs are evaluated autonomously for each transportation equipment, V ∈ Γπ, ∀π ∈ Πowner.

Parameters VC1V and VC2V are unitary variable cost issues. The former is a transportation cost coefficient proportional to the number of flows assigned to resource V, while the latter is a cost coefficient pondered over the whole set of compatible flows that can be assigned to that resource. - Contracted transportation capacities: fixed costs are due to a transportation structure maintenance, accordingly to the global capacity contracted, while variable costs depend on the transportation resource usage and assignment of transportation equipments to suitable flows.

- Processing tasks: for these events, the fixed and variable costs depend on the equipment usage.

In conclusion, the objective function (eq 25) and the constraints 1-24 define the MILP formulation developed for the SC detailed scheduling problem. This was implemented in the

GAMS language and solved using a standard branch-and-bound (B&B) procedure (CPLEX). 5. Industrial Example 5.1. Supply Chain Description. A supply chain (SC) is considered where different blends for civil construction purposes as well as for wood-related industries are produced on different facilities structures distributed over different country locations. The operational structure is characterized by four types of clusters: (i) supply, S1-S5; (ii), production, I1 and I2; (iii) customization, finishing, packing, and labeling operations, PL1 and PL2; and (iv) distribution, DC1-DC10. These are linked by a connectiVity network (Figures 1 and 6) involving 13 transportation structures (π ) 1, ..., 13) that guarantee the materials flows along the chain. Each SC site represents a chain partner that locally manages the operation of the installed resources. Sites S1, S2, and S3 provide three classes of raw chemicals (ChA, ChB, and ChC) to the plants, I1 and I2. Partners S4 and S5 supply both packing sites, PL1 and PL2, with the finishing additives, AdF and AdR, and with the required packing materials, P50 and P5 containers, that have a capacity of 50 and 5 volume units (u.v.), respectively. At plant I1, two preparing lines are installed, PP1 and PP2. These produce, respectively, the intermediate blend IB1, through the transformation of 55% ChA and 35% of ChB (mass percentages), and IB2, from the conversion of equal mass percentages of both raw chemicals, ChA and ChC. Instead, a single production line, PP3, is available at plant I2. This produces the intermediate blend IB2 using the same production recipe, defined for I1 partner. Six final blends, FB1-FB6, are produced from the prior intermediate blends (IB1 and IB2). These result from different customization processes of intermediate blends, performed at PL1 and PL2 sites. Each customization process involves a finishing task followed by a nonwaiting suitable packing and labeling task. Finishing tasks follow the same production recipe at both PL partners and involve the addition of different

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Figure 7. Schematic representation of the SC operational scenarios analyzed.

percentages of additive agents, AdF/AdR, respectively, into IB1 and IB2 blends. Each PL partner has two independent resource structures, PLS1/PLS2 at PL1 site and PLS3/PLS4 at PL2 partner, to perform the customization processes formerly described. PL1 site has the following: (i) a single customization, FB3, of IB2 blend (P5 containers), done at PLS1 structure and (ii) two feasible customizations, FB1 and FB2, of IB1 blend corresponding to different packing options (P50 and P5 containers), performed, respectively, at PLS1 and PLS2 processing structures. An equivalent situation with different customization decisions arises at PL2 site. There, two customizations, FB4 (P50) and FB6 (P5), of IB1 blend are performed at PLS3 and PLS4, respectively, and a single customization, FB5 (P50), of the concrete blend IB2, is done at PLS3. Further details on problem data are summarized and presented in Appendix A, Tables A1A5. Concerning the partnership operational decisions, each PL site fulfills a cluster of distribution sites that afterward ensure the requirements of final customers located on its neighborhood. PL1 partner supplies distribution sites, DC1-DC6, located in the north of Portugal, while PL2 supplies DC7-DC10 distribution partners located in the south. Also, the geographical positions fulfilled by each DC site (local customers clusters) have minimal weekly demand requirements as well as upper daily capacities, defined in here as the maximal daily demand (Table A2). All the SC sites have their own storage resources and may use both dedicated and sharing options. Different operational preconditions are defined at the materials storage, namely, (1) at the beginning of the scheduling horizon, each supply chain site has a minimal stock of every suitable product defined as 25% of the storage capacity dedicated to that material state; (2) at least the same stock levels must be left at the end of the time horizon to allow the starting of the next scheduling period; and (3) no storage levels less than 5% of the storage capacity dedicated to each material state are allowed, at any scheduling time. The connectivity network is globally characterized by 13 transportation structures where 56 transportation flows are defined to describe the materials’ mobility (see Figure 6). Each supply chain site may have in-house dedicated transport resources and it may also contract some transport capacity to a provider company (contracted structures). In the latter, a dedicated or sharing situation may arise between sites (Table A3). Transportation structures have a limited capacity, resulting from the set of transport resources defined within the structure, and its usage is limited to a defined set of suitable materials. Thus, because of the materials’ properties and customization options (bulk, packed, etc.), different transport decisions are

considered for compatible materials (AdF/AdR, P5/P50, FB1/FB3, or FB4/FB5) sharing a common transportation resource (Table A3). Apart from the material suitability and capacity requirements, minimal and maximal transportation capacities are considered. The minimal capacity request is defined as 10% of the equipment capacity, for raw chemicals and intermediate blends, and as 20% of the resource capacity for final blends. A full resource charge is allowed as an upper capacity bound. Finally, the economical data considerations are as follows: (i) Each material has a storage cost, SCs, and two market values, the sourcing or production price, pws, and the retailing price, pms ) pws(1 + ∆pws). The latter is an incremental percentage relative to the former, defined as 35% higher. (ii) Model events, production, packing, storage, and transport operations, have fixed and variable cost terms (Table A5). Having defined the case-study details, it is now important to identify the problem master goal. In here, the maximization of SC global profit is considered as the optimization objective. 5.2. Scheduling Operational Scenarios. In order to realize the impact of market opportunities and partners options on the achievable schedule plans and economical results, different operational scenarios were studied. Three operational scenarios are considered: (1) closed, (2) partially opened, and (3) totally opened operation; see Figure 7. The major characteristics of these scenarios can be summarized as follows: (1) Closedsonly the SC boundary clusters, sites defined within the supply and distribution clusters, are allowed to exchange materials with the outside market. Thus, no market opportunities are accounted for the remaining SC partners. (2) Partially openedsdifferent market options are attained by allowing production partners (I1 and I2 plants) to deliver their final products (not packed intermediates, IB1 and IB2) to external customers, within defined market capacity bounds. Every SC site can also decide on external material supplies, exchanges with external providers, done at a retailing price, pms. (3) Totally openedsextended market decisions are integrated by allowing SC sites to decide on both the material deliveries to external customers and the receipts from external providers, within defined market capacities and prices. Therefore, when moving from scenario 1 to scenario 2, the production and customization partners can decide on other material providers than the SC partners. Production sites are also allowed to deliver their final products (intermediate blends) to the outside market. These partners options are extended a bit far when moving to scenario 3 where customization sites can additionally deliver final blends to external customers. All the material exchanges are accounted for within feasible market bounds and predefined partnership conditions. These operational scenarios do not hold a complete partners rationality study since individual partner’s decisions, even allowed, are submitted to a master integrative goal, the SC global profit.

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Table 1. Economical Values Achieved for the Computed Operational Scenarios economic issue (euro/week) 1. weekly existences 2. delivers 3. transport 4. production and packing i. variable ii. fixed 5. external receipts 6. storage total cost issues total income issues global profit

closed

partially opened

opened

112061 1396144 70402

132942 1440299 68142

99521 1502400 65977

255463 94900 542702 29986 993453 1508205 514752

259711 96600 599259 31047 1054759 1573241 518482

261312 96600 625595 28306 1077790 1601921 524131

5.3. Optimal Scheduling Plans. The SC industrial problem was solved for the described operational scenarios, while considering a scheduling horizon of 5 days/working week with 8 h/day, for all the SC partners. A 2 h discretization interval has been used to describe the scheduling horizon. The maximization of the SC global profit is defined as the optimization objective. Because of the extensive scheduling results obtained, a single operational scenario is discussed (closed operation). However, some major remarks will be provided for the remaining two scenarios in order to enhance the impact of the studied market opportunities and partners’ options. The optimal schedule obtained for the closed scenario leads to a global profit of 514,752 euro/week; see Table 1. The costly operation is the external supply of materials (542 702 euro/ week). This accounts for the scheduled supplies of chemicals, additives and packing materials as well as for some nonprogrammed fulfillments done to S1 and S5 suppliers. As a global behavior, the transportation performance increase while saving at handling of stocks and traveling times (e.g., I1 and I2 plants are sourced by the geographically closer supplier, S1 and S3, respectively). Higher service levels are reached by deciding on some contracted transportation capacities, even when incurring on a more costly transportation decision. As an example, supplier S1 improves the service levels to plants by choosing the train option (contracted structure, π)1). A similar situation is observed between plants and PL partners. There, plant I1 requires frequently a contracted transportation structure, π ) 8, to hold on the service levels to PL sites. Although, contracted capacities are not a decision subject to some SC sites. This is observed namely at S2 and S3 partners that are fully dependent on contracted transport capacities and thus economical and operational decisions result highly constrained. The economical balance of transport operations (70 402 euro/ week) accounts for both fixed and variable costs. Because of

the capacity conditions and profitability issues, the transportation resources operate usually at their upper capacity limits, especially when the transport decisions require costly capacities (contracted structures). At this operational scenario, the SC global performance is fully dependent on the transportation efficiency since external market options are strongly constrained. The observed transportation requirements justify, almost all the time, a full equipment charge (full resource capacity), and therefore, resource sharing is not a relevant working practice. However, the geographical dispersion of the DC sites, combined with the due date requirements, enforces some final product transports not justifying the whole equipment charge, and some unused capacity is then observed (Figures 8 and 9). Another important profit reduction results from the economical evaluation of processing (production and customization) operations (350,362 euro/week). These account for both variable (255,462 euro/week) and fixed (94,900 euro/week) operational costs. The available structures operate at their upper capacity limits during almost all the week, since a profitable operational policy is achieved by splitting the variable costs among higher batch dimensions; see Figure 10. The global performance observed suggests a very constrained production (reaching the minimal demand levels) for the lessprofitable final blends, FB3 within PL1 and FB4 within PL2. This arises because of the competition for (i) a specific packing structure, PLS1 at PL1 site and PLS4 at PL2 partner (between FB1/FB3 and FB4/FB5) and for (ii) an intermediate blend, IB1 at PL1 partner and IB2 at PL2 (e.g., FB1/FB2 at PL1 and FB5/ FB6 at PL2). The storage costs (29,985 euro/week) summed up with the production counterpart are mainly responsible for the lower storage levels observed around every SC partner during the week. Almost all the material states exhibit a storage profile not far from the defined safety levels, and even at the end of the schedule horizon, only a few materials have an inventory level greater than that defined as a scheduling final requirement (inventory at the end of scheduling horizon). Therefore, the profit incomes achieved by the material assets (i.e., materials inventory due to the week operation, ΣSsH - Ss0) are low (112,061 euro/week) (Table 1). Finally, the delivers of final blends into the final costumers or aggregated positions are mainly responsible for the global SC profit achieved. These are held by the minimal distribution requirements for the tight due dates and less-profitable final blends and by the production and transportation performances achieved at this constrained operational scenario.

Figure 8. Operational scheduling plan for transportation structures connecting PL1 site with the distribution sites, DC1-DC6, while considering a closed scenario.

Ind. Eng. Chem. Res., Vol. 47, No. 1, 2008 127

Figure 9. Operational scheduling plan for the transportation structures connecting PL2 site with distribution sites, DC7-DC10, while considering a closed market scenario.

Figure 10. Operational scheduling of processing resources, while considering a closed market scenario.

The scheduling plan achieved involves a complex and combined set of cost and capacity decisions. These can be summarized as follows: • Process (production and customization): transformation resources are typically used at their upper capacity limit to enhance profitable blends. • Transportation: larger assignment of inner transport structures to satisfy demand requirements, while saving on transportation costs. Contracted structures are commonly utilized to enhance higher service levels and to fulfill some nonreached minimal demands requests. • DeliVeries: these are the major profit incomes since they balance the economical returns resulting from the material exchanges going to customers and prevent the increase of storage expenses. • Storage: almost all the material states reveal a storage profile not far from the defined safety levels, and even at the end of the schedule horizon, only a few materials have an inventory level greater than that defined as a final scheduling requirement (25% of the capacity dedicated to each storable material state). 5.4. Remarks on Global Schedule Scenarios. Having discussed the scheduling details for scenario 1 (closed operation), it is now important to analyze how the consideration of different partners and market options modifies the prior schedule plan. This analysis will be performed concerning the

major management decisions involved around each operational scenario and their impact on the operational schedules obtained. Therefore, the following considerations can be performed on the partially opened scenario while being compared with the previously discussed (closed) scenario: • Receipts: due to the new sourcing conditions (external providers opportunity), higher, while costly, service levels were achieved at plants and PL partners. • Production and customization: different production and customization schedules were obtained as a result of the materials availability at plants and PL sites. A costly operation (at both fixed and variable cost issues) results at the partially opened scenario. Variables costs increase more than fixed and, thus, a profitable resource occupation results. • Stocks: the storage costs increase slightly while moving from closed to partially opened scenario, but not as much as the week existences; see Table 1 (issues 1 and 6). This suggests that the stock increment occurs at a time instance near the final schedule horizon in order to allow an increase of the material existences (issue 1) without incurring important storage costs (issue 6); • External deliVeries: higher incomes were reached as a result of the opened market opportunities (deliveries of IB blends). These also allow the cheapest transportation schedule. When the market constraints imposed to the final blends distribution are released (opened operation), the material flows

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Table 2. Computational Statistics Resulting for the Computed Operational Scenarios model statistics for the three studied scenarios no. of variable no. of integer variables no. of constraints optimal integer solution, OIS ∆PO-C ) 3730; ∆O-PO ) 5648; ∆O-C ) 9378 best integer solution possible, BISP ∆PO-C ) 13 868; ∆O-PO ) 6 546; ∆O-C ) 21 699

14 085 9 944 34 130

operational scenario

closed

partially opened

opened

no. of nodes % optimality, RGa CPU s

15 226 2.07 5 286 514 752

7 387 4.04 2 836 518 482

4 017 4.14 1 489 524 130

525 415

539 283

545 829

a The % of optimality is defined by the relative gap, RG: RG ) [(BISP - OIS)/OIS] × 100%, where BISP is the best integer solution possible and OIS is the optimal integer solution satisfying the tolerance defined.

to the external market increase and a global profitable operation (524,130 euro/week) is reached at this operational scenario. Themajor remarks on this operational scenario can be compared with the prior schedules presented as follows: • Process: production and customization performances are improved since the same fixed costs (96,600 euro/week) were obtained for higher batch dimensions (variable costs ) 261,312 euro/week). Therefore, the production costs of the opened scenario are higher but distributed among larger amounts of products, and accordingly, less costly blends are obtained (Table 1). • Transportation: as a result of the new market opportunities, a global transport reschedule is observed. The transportation costs decrease (65,977 euro/week against 68,142 euro/week at partially opened or 70,402 euro/week at the closed scenario) because of the larger material deliveries globally observed. • DeliVeries: DC partners are satisfied at their minimal demand levels since a profitable operation is reached by allowing PL partners to decide on other market opportunities (final blends are partially outsourced to local costumers instead of being transported to the DC partners). The outsourced amounts depend on the profitability margin raised by each blend production and customization. These go on from 25% of the customized FB3 blend till 47.5% of the FB5 final blend obtained. • Storage: some savings on storage costs (28,305 euro/week) were reached at the opened scenario. An improved capacity utilization is observed, and thus, a less-expensive storage schedule is obtained. As was formerly noticed, all the market decisions considered (receipts and deliveries) were defined according to the customers demand requirements, daily capacity absorptions, and due dates. In conclusion, concerning the operational perspective, scenario 1 (closed operation) is the one that produces the lower amounts of intermediate and final blends, but at the same time, it is responsible for the larger transportation flows from suppliers to DC site positions. On the opposite side, scenario 3 (opened operation) is the profitable one. The integration of new partners and market opportunities improves production and packing rates while savings on transportation and storage costs. The observed results advise on the economical and managing benefits that can be raised through the integration of different partners options, within feasible operational limits and market bounds. Finally, in terms of model statistics, Table 2 reports the model dimension obtained for the three discussed scenarios. The models were solved using the GAMS package coupled with the CPLEX solver version 6.6.1 in a Pentium III. The optimal solutions were obtained for the LPs and CPU times reported while considering a maximal relative gap of 5%. The relative gap increases from the least to the most profitable scenario (from closed to opened scenario), and the optimal solution obtained for the opened scenario is the most deviated from the best integer solution possible, BISP. According to this observation, the differences obtained between optimal integer

solutions could be larger if the models were solved for the smallest relative gap. 6. Conclusions The inherent complexity typically found in supply chain structures requires detailed and robust models to handle all the requirements properly and effectively. The proposed approach provides the optimal scheduling of supply chains while combining, at a single-level formulation, the structural configuration with different operational conditions and market decisions. Supply chain topology, operability, and processing precondition requirements are explicitly taken into account. Also, different transportation decisions, market opportunities (providers and customers), and partnership relations were combined within a master integrated scheduling objective. Therefore, some advantages can be taken from its implementation at an industrial level. Different management options, not accomplished at a strategic level, can be exploited to anticipate industrial decisions. To reach this scope, different operational conditions and market opportunities were explicitly considered at the detailed SC operation. A set of operational scenarios, representing different management decisions and operational conditions, were solved and discussed. As a final result, the model provides a detailed operational plan at the production, storage, and transportation levels. The assignment of operations, production, customization, storage, and transport as well as the allocation of resources and materials is fully identified while satisfying predefined market requirements and guaranteeing a maximum global profit. A real case study representing a countrywide industrial SC was solved, and good results were obtained within a reasonable margin of optimality. Some model adjustments can be integrated to account for international partnership options not accomplished at the industrial example discussed. The drawback of the high generality of the proposed formulation is that the resulting MILPs may become hard to be solved. This is justified by the huge number of variables with an integer domain as well as by the large number of hard constraints that have to be satisfied (as incompatibility relations). As future work, the authors are exploring the model efficiency through some auxiliary model developments such as logical constraints and cutting plans, among others. Furthermore, and considering the model generality, some formulation developments concerning the integration of different partners’ rationality decisions at the operational level are being studied. These require wide market analysis, namely, international market decisions accounted by the integration of different price options and market bounds. Also, different transportation network options, as reverse logistic integration, are being studied in order to get further operational improvements concerning the global balance of material flows along the chain.

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Appendix A: Data of the Industrial Example The presented data are organized as follows: (1) partners operation, Table A1 (tasks processing times, pti, resources capacities, CJj, and suitability, Jij); (2) transportation structures, Table A2 (capacity and transportation suitability, for inner (π ) 2, 5, 6, 7, 8, 10, 11, and 12) and contracted structures (π ) 1, 3, 4, 9, and 13); (3) market options, Table A3 (customers’ minimal weekly demands, Min W, and maximal daily capacities, Max D); (4) economical data, Tables A4 and A5, for materials (production prices and storage costs, Table A4) and for SC operations (tasks and flows, Table A5).

Table A1. Supply Chain Processing Resource Descriptiona I1

I2

site (operational partner) resource, j ∈ JP

PP1

PP2

PP3

CJ/j

12 500/16 200 i1 4

4 250/4 200 i2 2

16 500/16 200 i3 4

capacity, suitability, JIj processing time, pti, h

PL1

a

PL2

site (operational partner) resource, j ∈JP

PLS1

PLS2

PLS3

PLS4

capacity, CJj a suitability, JIj processing time, pti, h

16 000/16 200 i4, i6 4, 4

3 200/4 200 i5 2

4 000/4 200 i9 2

16 000/16 200 i7, i8 4, 4

u.v./u.m. ) volume, l, mass unit, kg.

Table A2. Supply Chain Transportation Structures Characteristics suppliers S1-S3

π ) 1 contracted

π ) 2 in-house

π ) 3 contracted

π ) 4 contracted

resource, V capacity, CVV

V1, V2 3500, 3500m

V3, V4 2000, 3000m

V5, V6 4000, 6000V

V7, V8, V9 2500, 2500, 4000m

supplier-plant S4,S5-I1,I2

π ) 5 in-house

π ) 6 in-house

π ) 7 in-house

π ) 8 contracted

π ) 9 contracted

resource, V capacity, CVV a

V10, V11 80, 100 AdF

V12, V13, V14 250, 250, 400 P50

V15, V16, V17 3500, 3500, 5000m

V18, V19 4000, 5000m

V20, V21 5000, 6000m

tomization PL1 and PL2 resource, V capacity, CVV b a

π ) 10 in-house

π ) 11 in-house

π ) 12 in-house

π ) 13 contracted

V22, V23, V24, V25 40, 60, 60, 80 FB1

V26, V27 600, 800 FB2

V28, V29, V30 75, 75, 100 FB4

V31, V32 900, 1200 FB6

Volume units: P50 ) 8P5, 1AdR ) 2AdF. b FB containers (weigh), 1FB1 ) 1.25FB3, 1FB4 ) 1.20FB5.

Table A3. Distribution Sites Characteristics and Market Demand Requirements site (op. partner) storage space capacity CJj P50/P5a demand Min W/ Max Db FB1 FB2 FB3 Max days/week

site (op. partner) storage space capacity CJj P50/P5a demand Min W/ Max Db FB4 FB5 FB6 Max days/week a

DC2 62.5 m3

DC1 100 m3

DC3 50 m3

DC4 62.5 m3

DC5 62.5 m3

2000/16000

1250/10000

1000/8000

1250/10000

1250/10000

250/90 2000/900 250/60 5

120/60 1800/900 150/80 3 (Mo/Wd/Fr)

120/100 1200/1000 100/100 2 (Tu/Th)

180/90 1200/600 150/75 3 (Mo/Wd/F)

200/90 2000/900 5

DC6 30 m3 600

250/250 2 (Tu/Th)

DC8 62.5 m3

DC9 75 m3

DC10 100 m3

DC10 100 m3

1 250/10 000

1 500/12 000

2 000/16 000

2 000/16 000

150/100 150/100 1 200/600 3 (Mo/Wd/Fr)

250/100 250/100 2 000/600 5

360/250 3 600/2 400 3 (Mo/Wd/Fr)

500/450 3 000/2 400 2 (Tu/Th)

P5 containers use ∼80% of the volumetric space unit (1P50 ≡ 8P5). b W ) weekly and D ) daily.

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Ind. Eng. Chem. Res., Vol. 47, No. 1, 2008

Table A4. Production Price and Storage Cost for the Material States site Si\state prod. price, pws SCs

(10-3

euro/u.m. day)

FSCj (euro/st*‚week) site\state

S1\ChA

ChB

S2\ChB

ChC

S3\ChA

ChC

S4\AdF

AdR

S5\P50L

P5L

0.42

0.28

0.50

0.75

0.50

0.75

34.5

134.5

1.25

1.75

2.52

1.68

1.68

4.20

2.52

4.20

0.207

0.816

0.0111

0.144

180

120

200

200

200

200

300.

200.

240

560

I1\IB1

IB2

I2\

IB2

PL1\FB1

FB2

FB3

PL2\FB4

FB5

FB6

prod. price, pws

0.90

1.50

1.50

175

18

180

175

180

20

SCs (10-3 euro/u.m. day)

5.40

9.00

9.00

1.05

0.108

1.08

1.05

1.08

0.108

FSCj (euro/st*‚week)

280

120

300

125

50

100

120

100

60

site DCi\state

DC1\FB1

FB2

FB3

DC2\FB1

FB2

FB3

DC3\FB1

FB2

FB3

DC4\FB1

SCs (euro/u.m. day)

1.05

0.108

1.08

1.05

0.108

1.08

1.05

0.108

1.08

1.05

FSCj (euro/st*.week)

126

105

19

84

126

140

108

102

90

102

site DCi\state

DC7\FB4

FB5

FB6

DC8\FB4

FB5

FB6

DC9\FB5

FB6

DC10\FB5

FB6

SCs (euro/u.m. day)

1.05

1.08

0.108

1.05

1.08

0.108

1.08

0.12

1.08

0.12

FSCj (euro/st*.week)

128

128

144

144

144

112

160

240

240

160

FB2

FB3 DC5\FB1

0.108 1.08 115

102

1.05 144

FB2

DC6/ FB3

0.108 1.08 156

250

Table A5. Fixed and Variable Costs for the Supply Chain Events (Tasks and Flows)a variable cost (euro/u.m.)

fixed cost structure j ∈ JP, i ∈ I (euro)

structure j ∈ JP, i ∈ I

fixed cost (euro)

variable cost (euro/u.m.)

structure j ∈ JP, i ∈ I

fixed cost (euro)

variable cost (euro/u.m.)

I1 PP1 1

1200

0.18

I1 PP2 2

850

0.32

I2 PP3 3

1800

PL1 PLS1

1700

12.5

PL1 PLS2 5

300

2.0

PL2 PLS4

1500

18.5

4 and 6

2100

20.0

PL2 PLS3 9

400

2.4

7 and 8

1750

17.5

structure π

fixed cost variable cost (euro) (euro/ch*‚∆t)

structure π

fixed cost (euro)

variable cost structure π (euro/ch*‚∆t)

fixed cost (euro)

variable cost (euro/ch*‚∆t)

175, 175

π)6 v12, V13 , V14

48, 48, 60 24, 24, 30

π ) 10 V22, V23, V24, V25

56, 64, 64, 80 28, 32, 32, 40

35, 43.75

π)7 V15, V16, V17

60, 60, 72 53, 53, 63

π ) 11 V26, V27

72, 84

36, 73.5

π)3 V5, V6

200, 275

π)8 V18, V19

160, 200

π ) 12 V28, V29, V30

72, 84, 84

36, 42, 42

π)4 V7, V8 ,V9

125, 125, 160 π ) 9 V20, V21

190, 220

π ) 13 V31, V32

π)1 V1, V2 π)2 V3 , V4

40, 50

π)5 V10, V11

52, 60

0.34

160, 200

26, 30

a For in-house structures, each transported material is balanced as a charge (to account for loading and handling costs), while for contracted structures, those costs are implicitly contracted per travel or charge.

Composed Sets

Appendix B: Nomenclature Indices i ) 1, ..., NT ) processing tasks j ) 1, ..., NJ ) processing resources l ) 1, ..., NF ) transportation flows s ) 1, ..., NS ) material states V ) 1, ..., NV ) transportation resource π ) 1, ..., NΠ ) transportation structures t ) 1, ..., H + 1 ) scheduling times

set Fπ Ici JIj KSj

Simple Sets (or Structural Sets) set element cardinal details

I IB ⊆ I

J JP ∪ JS

i j NT NJ tasks/resources

∏

L l NL

(∏owner ⊆ ∏)

V

π V N∏ NV flows/structures/ resources

S (S′ ⊆ S)a s NS material states

out Sin i /Si out SIin s /SIs out SLin s /SLs Γπ

a

Set S considers all the SC material states while S′ accounts for storable materials, including the material states to be delivered (SD) and receipts (SR, involves raw materials, Sraw, and other states, SR\Sraw) from/into the SC sites.

¥πl Λπ

details {element ∈ set (such that): condition} ; |set| (if required) {l ∈ L: flow l is defined through transportation structure π } {j ∈ JP\JB: processing resource j is suitable to perform task i ∈ I} {i ∈ I: processing task i can be performed at resource j ∈ J} {s ∈ S′: material state s can be stored in resource j ∈ JS} {s ∈ S: material state s is the input/output of processing task i} {i ∈ I: processing task i produces/ consumes material into/from state s} {l ∈ L: flow l gets/releases material from/into state s} {V ∈ V: transport resource V belongs to transportation structure π} {l′ ∈ Fπ: flow l′ is compatible with flow l}; |¥ πl| ) ϑπl {¥πl ∪ {l}, ∀l ∈ Fπ: set of compatible flows’ sets defined in structure π}

Ind. Eng. Chem. Res., Vol. 47, No. 1, 2008 131

Continuous/Integer Variables Dst ) amount of material state s delivered to external customers, at the beginning of time interval t QΛlt ) global amount of material transported by flow l, using a set of suitable transportation resources, at the beginning of time interval t QλVlt ) amount of material transported by flow l, while assigned to resource V, at the beginning of time interval t and during ttl/∆t time intervals Qpijt ) amount of material processed by task i, using resource j, at the beginning of time interval t and during pti/∆t time intervals Rst ) amount of material state s received from the outside market into a supply chain site Sst ) amount of material state s (a material in a given location) available at the beginning of interval t Binary Variables YλVlt )

{

1 if processing task i is assigned to equipment j, at the begining of time t 0 otherwise

Ypijt )

YνVt )

{

YDst )

{

1 if transport flow l is assigned to transport equipment V, at the begining of time t 0 otherwise

1 if transport equipment V is occupied at time t, with the transport of some material(s) 0 otherwise

{

1 if deliver of material state s is performed by customers at time t 0 otherwise

Time and Capacity Parameters CJj ) capacity of the processing resource j (volume, mass, rate, etc.) CVV ) capacity of transport resource V (volume, mass, etc.) Up DLow st /Dst ) minimal/maximal amount of material state s that can be delivered to external customers, at the beginning of time interval t D ˆ Low ˆ Up st /D st ) minimal/maximal amount of material state s to be delivered, by inner transportation resources, to external customers, at the beginning of time interval t pti ) processing time of task i represents the time required to fully execute the processing operation QChs ) fixed capacity defined to a nonschedule charge of material state s (amount/charge) Up RLow st /Rst ) minimal/ maximal amount of material state s that can be sourced, by an external provider, at time t. These operations usually involve contract periods dependent on the sourced material state, TRs ttl ) transportation time of flow l represents the travel time or duration of the transport operation Φmin /Φmax ) minimal/maximal utilization factor of the ij ij processing resource j, by task i max φmin ) minimal/maximal proportion of storage resj /φsj source j capacity, dedicated to material state s out Rin is /Ris ) rate or proportion of state s undergoing/ leaving task i, accordingly with the task recipe Ξπ ) max∀l∈Fπ ϑlπ + 1, maximum number of compatible flows defined in structure π

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ReceiVed for reView February 19, 2007 ReVised manuscript receiVed August 30, 2007 Accepted September 7, 2007 IE070262A