Supply Chains Planning with Reverse Flows: Optimal Alternative Time

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Supply Chains Planning with Reverse Flows: Optimal Alternative Time Formulations Ana C. S. Amaro†,‡ and Ana P. F. D. Barbosa-Povoa‡,* † ‡

Instituto Superior de Contabilidade e Administrac-~ao, ISCAC, Quinta Agrícola, 3040 Coimbra, Portugal Centro de Estudos de Gest~ao do IST (CEG-IST), Instituto Superior Tecnico, Universidade Tecnica de Lisboa, Av Rovisco Pais, 1049-001 Lisboa Codex, Portugal ABSTRACT: In this paper the optimal planning of supply chains is studied. Two alternative formulations are developed to model the supply chain optimal planning: a discrete and a continuous-time formulation. The former considers the uniform split of the planning horizon into equal time intervals while the continuous-time counterpart involves the definition of a set of time slots, of unknown duration. Each slot dimension is optimized simultaneously with the planning events. Both approaches account explicitly for the integration of topological, operational, and market supply demand constraints and requirements. The supply of final products (forward flows) and the return of nonconform products (reverse flows) are simultaneously coordinated at the planning level of the supply chain optimization. The proposed formulations result into mixed integer linear programming (MILP’s) models. A detailed plan is obtained for each formulation approach, which improves the supply chain operability by exploiting general resource capacities (e.g., transforming, storage, and transportation) and resource sharing policies based on the suitability of equipment/tasks, economical performances, and operational conditions. The applicability of the proposed formulations is illustrated through the solution of an industry-oriented case study.

1. INTRODUCTION Presently enterprises are facing important and continuous management challenges emerging namely from the technological, economical, and social alterations that characterize nowaday global markets. These introduce a clear change of paradigm on management strategies. The global-wide international market scale, the aggressive competition, the demand variability, the requirements of new products with short life cycles, and the huge variety of supply alternatives for similar products and services are some explanatory reasons to the observed management challenges. These issues are hard or even impossible to handle by single industrial positions in a profitable way. Indeed a single company can rarely control the production of a commodity together with sourcing, distribution, and retail in global-wide markets.1 Thus, enterprises must concentrate on the achievement of collaborative strategies supported by shared policies between members (partners’ relationship) in order to enhance, in a systematic way, global coordinated solutions, at different decision levels.2 This integration perspective introduces novel operational market and business concepts (cross functional and business coordination), scales, and requirements and defines the foundations of the supply chain (SC) structures. In this context, multienterprise networks, global enterprise planning, coordination, cooperation, and responsiveness to customers, at global and local levels, are gaining increasing attention and have been recognized as critical for ensuring effectiveness and business competitiveness sustainability as well as growth.1,3 Accordingly, supply chain management (SCM) strategies and tools are capturing an increasing interest from both academics and industrial practitioners.4 In recent years, and following this r 2011 American Chemical Society

motivation, there has been a multitude of efforts focused on providing improvements of SC management and optimization.5 These have been mainly concerned with the design aspects of SC structures, where in some cases also planning aspects were taken into account.6 Two important research strategies were followed, a stochastic and a deterministic approach. The former, results frequently into complex nonlinear optimization problems requiring strong assumptions to achieve an optimal or near-optimal solution.7,8 On the other hand, the deterministic approach9-12 results typically into large mixed integer linear problems based essentially in a discrete time approach to represent the time domain.13-15 Moreover, some linear formulations based on continuous-time representations have also been proposed16-18 for different SC management problems. Independently of the research approaches considered, it has been widely recognized that effective management of SC is a complex task. The implementation of several formulations proposed so far result into problems that are hard or even impossible to solve within a reasonable margin of optimality. Furthermore, few studies1,19,20 have considered the integration of the different SC activities.21,22 This is typically the case of the SC planning problem where there is a lack of generic models that are able to account for the supply chain global characteristics.1,23 In addition, new challenges emerge from the recognition of environmental concerns and from the requirements of sustainable Special Issue: Puigjaner Issue Received: July 2, 2010 Accepted: December 20, 2010 Revised: December 12, 2010 Published: March 25, 2011 5005

dx.doi.org/10.1021/ie101416q | Ind. Eng. Chem. Res. 2011, 50, 5005–5022

Industrial & Engineering Chemistry Research

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Figure 1. Closed-loop supply chain (CLSC).

development.24,25 Thus, managing strategies based on the control of the forward material flows and on the feedback information flows are not enough to ensure global chain performances nor profit purposes. Often reverse flows of nonconform materials26 (reverse logistics) are strongly engaged with the provider’s feasibility (producers and transportation capacity) and cannot be economical nor operationally disregarded. These new aspects lead to even more complex SC structures that require the usage of decision supporting tools to help the decision making associated process. The integration of these reverse flows going from customers back to the producers, where recovery or remanufacturing operations are performed, resulted in the appearance of the closed loop supply chains (CLSC).27,28 Following the academic and industrial motivations previously identified, the major scope of the proposed study is to develop a modeling framework to help the decision making process, at the SC planning level, while considering market and economical characteristics of SC partners. This paper integrates operational issues, as materials transportation and recovery operations, within a collaborative SC structure. A centralized managing strategy defined over a multiperiod planning problem is modeled. Two alternative model formulations are developed. The first one was initially proposed by Amaro and Barbosa-P ovoa13 and it involves a discrete time representation, while the second one is presented along this paper and uses a continuous-time definition. Novel concepts are introduced to allow the complete representation of supply chain structure and its operational characteristics while keeping the linearity of the proposed mathematical formulations. The resulting MILP problems are solved using a standard branch and bound (B&B) procedure. The performance of the developed models and the events allocation obtained within each time representation are compared. This comparison is performed based on the application of both approaches to the solution of a real case-study.

2. SUPPLY-CHAIN CHARACTERIZATION In this paper, we look into a generic industrial supply chain where reverse flows are accounted for (see Figure 1). This involves an integrated superstructure of partners such as suppliers, industrial facilities, transportation providers, distribution, customers, and disposal sites, geographically distributed and having a collaborative relationship, Figure 1.

The major difference between traditional SCs and CLSCs results from the consideration of reverse logistic issues representing the possibility of handling the return of materials from customer positions back into suppliers or manufactures (Salema et al.27). In the considered structure these reverse flows are present resulting into a closed-loop structure (Figure 1). Within such structure a wide range of products can be manufactured at the existent industrial facilities which may use dedicated or multipurpose processes. The products are distributed to customers (i.e., marketplaces or aggregated market regions) through distribution sites that represent centralized storage positions. Nonconform products are dealt in an integrated way and collected from the costumers or distribution sites and (1) sent to disposal sites which can be seen as incineration entities where these products are processed or (2) recovered at the manufacturing sites or even at some distribution sites (depending on the level of nonconformity) and introduced again into the chain. Finally, transportation providers guarantee the material flows among SC partners and into/from customers’ positions.

3. MODELING DETAILS AND CONCEPTS Planning problems are classified at a tactical management level and involve typically a medium to a large time horizon, Figure 2. An important issue to be considered when dealing with such problems is the trade-off between the model dimension, arising intrinsically from the time horizon duration, and the optimality requirements’ expected at the planning solution. Usually, the time requirements associated with the execution of SC operations (e.g., process, transportation) are smaller (e.g., hours, days, etc.) when compared to the planning horizon (weeks, months, etc.). Thus, large dimensional models are often obtained if detailed time grids are used to fully support the description of the SC events. Therefore special care should be given to the establishment of the correct trade-off between the planning horizon period and the duration of the planning events considered. A particular emphasis has to be placed on the time representation chosen in order to account for the SC operational details while keeping a controlled model dimension. Furthermore, the approach adopted to represent the time can be particularly important to achieve the desired equilibrium between the model implementation and its solution feasibility as 5006

dx.doi.org/10.1021/ie101416q |Ind. Eng. Chem. Res. 2011, 50, 5005–5022


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Figure 2. Supply-chain management levels.

Table 1. Time Feasibility Conditions working plan

modeling flexibility

I. Non Interrupted Program (Continuous sequence of events);

a priori unconstrained time domain

I. non-pre-emptive operations

a priori constrained time domain

a priori constrained time domain

II. pre-emptive operations

a priori unconstrained time domain

partners’ operability

modeling flexibility

upper and lower time bounds II. Interrupted Program (regular/nonregular interruptions)

well as the right balance between the model detail and the solution performance. Given these issues, the main objective of the present paper is to develop a planning formulation that accounts for supply chain global coordination and dependencies, while keeping the model generality and the solution optimality, the latter within an acceptable relative gap. To achieve this, it is important to identify a time representation that enables the description of the main planning characteristics with the lowest dimensional requirements. This is here performed through the implementation of a modeling procedure based on the possibility of events’ replication (in the sense of repetition or copying of events) that ensures the optimality of the planning solution while observing its operational feasibility and avoiding a large aggregation of supply chain entities (e.g., general events, resources, and materials). This procedure is implemented considering two different approaches to model time within the planning formulation: a discrete and a continuous time representation. 3.1. Characterization of Planning Horizon. The planning horizon representation considers a macroscale description of the planning time domain. The main concepts involved at the planning master scale are related to the identification and characterization of both the SC working plan and the operational policies associated with the installed processes. The former is concerned with the feasibility conditions related to the working practice of the SC partners, while the second one considers the operational requirements related to the modeling of general SC events (e.g., transformation, supply, delivery, etc.) and instances (e.g., equipments, facility structures, etc.). These can introduce different a priori conditions into the model formulation and are called time feasibility conditions (Table 1). On the working plan, two major concepts are considered: (i) interrupted and (ii) noninterrupted program. The latter represents a 24 h time table of operation with no weekly or monthly regular interruptions. The planning horizon, Hp, is a continuous time domain, with two prior time bounds, a lower (Ht-1) and an upper bound (Ht), Figure 3. The lower bound is set to zero, if no planning period is preceding the former period of the actual planning horizon, otherwise it

Figure 3. Planning horizon with regular or periodic interruptions.

assumes the value corresponding to the upper bound of the prior planning period. Instead, if some regular or nonregular interruptions need to be considered, the time domain results into a discrete set of continuous time intervals (subset of time domains, Figure 3). The concept of a regular interruption is used to characterize all the periodic planning interruptions (e.g., weekend stops) while the nonregular interruption concept reports any nonperiodic stops known in advance (e.g., a priori known interruptions as holidays, setups, etc.) (Figure 3). Independently of the working plan considered, the planning horizon is characterized by the same upper and lower time bounds, Ht and Ht-1, respectively, but different internal time domains can occur. At the noninterrupted working plan, the time horizon is internally unconstrained and any time split (that fulfils the planning horizon bounds, Ht and Ht-1) can be implemented to account for the required planning details. These working plans do not introduce a priori time requests and therefore the feasibility conditions are intrinsically ensured (Table 1). For the interrupted working plans some intrinsic time limits exist. These are accounted through the integration of specific time bounds within the planning horizon, which split the time domain into a set of regular or nonregular time intervals. In both cases, the planning horizon is represented by a discrete set of continuous time intervals, each one representing a planning period. For regular planning periods (i.e., equally spaced time intervals) a macro time grid is generated to represent the time feasibility 5007


Industrial & Engineering Chemistry Research conditions and a time unit, corresponding to the grid dimension (interval duration), which can be considered to move between successive planning periods (i.e., master time bounds). If a nonregular set of interruptions has to be considered a nonuniform time split may be required to define the time bounds generated by those irregular interruptions. In both cases, the planning horizon is defined as an a priori constrained time domain (Table 1). Another aspect involved in the characterization of the planning horizon is the operational policy. Depending on the processes installed at the SC partners, different operational conditions can occur. These are related to the involved operations, its suitability as well as with general resource requirements. Two major concepts are introduced to classify the partners’ operability concerning the set of suitable operations: (i) preemptive and (ii) non-pre-emptive operations (Figure 4). The pre-emptive mode considers all the operations that can start at a given time (macrotime point), be interrupted, and continued later, until its full completion (e.g., a mixing operation, task1, started at the beginning of the week, interrupted to execute another mixing operation, task2, and finished after the weekend, Figure 4). On the other hand, the non-pre-emptive mode considers all the operations (general events) that must start and finish within defined time bounds (e.g., a batch operation, task3 or task4, should start and be finished within the week duration since no weekend crossover is allowed). The non-pre-emptive operability is then more constraining than the pre-emptive counterparts. Depending on the time scale details to be implemented, the integration of non pre-emptive operability may require the consideration of further feasibility conditions into the planning time horizon. As an illustrative example consider that we wish to make an 8 hours daily plan and there is a chain partner having some batch processes that take at least 10 h. If those batch tasks are non-pre-emptive operations, the consideration of that daily macrotime unit would not be feasible. Moreover, other time requirements such as supply time tables and delivery times among others should also be considered within the planning horizon and observed while defining the time scale. As it was previously illustrated for non-pre-emptive operations, an equally infeasible situation arises if the planning time bounds are larger than the duration between regular receipts or delivery events (e.g., raw material receipts every 3 days). In this case it is necessary to decide on the aggregation of those events within the defined planning bounds or otherwise develop time bounds consistent with the occurrence of those events. Thus, depending on the details required to represent the

Figure 4. Pre-emptive (task1 and task2) and non-pre-emptive (task3 and task4) mode of operation.

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time domain, different feasibility conditions have to be observed. Some combined analysis of the working plan, the operations policy, and the time requirements of general events is fundamental to decide on time scale details. This set of time conditions is usually integrated in the modeling approaches as global time feasibility bounds and are classified as a priori or necessary time conditions. In this paper, and as referred above, two modeling approaches are considered: the discrete and the continuous time approach. Also, both types of operational conditions are modeled (i.e., preemptive and non-pre-emptive operations) as well as different supply and delivering conditions. 3.1.1. Discrete and Continuous Time Representations. At the discrete representation, the planning horizon (Hp) is split into a set of uniform (equally spaced) intervals {ΔPp = ΔP ⊂ IRþ, p ∈ Ρ P ⊂ IN: p∈ΡΔPp = Hp} and a set of aggregated time instances, called macrotime units are obtained (e.g., a 3 months planning horizon, Hp, divided into a set of weekly intervals, ΔPp = ΔP). Within each macrotime unit a non-pre-emptive mode of operation is considered that describes the time tables of each supply chain partner where once an operation has started it cannot be interrupted, Figure 5. On the other hand, at the continuous-time representation a different approach is considered, Figure 6. The time horizon is divided into several intervals (slots, R), having an a priori unknown duration that will be determined by the optimization procedure simultaneously with the events assignment. Depending on the planning problem, different operational conditions and details can be required to fully represent the time feasibility conditions. The time horizon can be an a priori unconstrained domain or otherwise some specific time points representing planning conditions may be used to represent the time feasibility requirements (Figure 6a,b). These, can represent different time conditions imposed namely by the material receipts, deliveries, or other operational requirements that introduce bounds into the a priori unconstrained continuous time domain. In Figure 6a no event is allowed to crossover the defined periodical time bound while in Figure 6b events are allowed to crossover those time bounds.

Figure 6. Continuous-time representation.

Figure 5. Discrete time representation. 5008


Industrial & Engineering Chemistry Research 3.2. General Events and Resources. Having described the modeling approaches used to represent the time domain it is now important to characterize some details on events and resources. Commonly completion of events (i.e., the time required to fully execute the event) is significantly smaller than the planning horizon. Thus a detailed description of events at the planning decision level often results in large dimension models. As referred above a strategy to balance the models’ dimension and the details of its representation is then crucial. Different procedures can be implemented to establish this necessary trade-off. The development of aggregation procedures is the main strategy used to represent the SC planning problems. These procedures can be achieved using different clustering subjects and scales, namely, by grouping specific sets of events (operational cluster), by allowing the replication of events within any subset of the time domain while keeping the details on events description (replication procedure), or alternatively by using both procedures or any combination of other clustering and/or aggregation subjects. In general terms, the main purpose of the aggregation procedures is to achieve an event description consistent with the details of the planning time domain. Thus, different events aggregation (clustering scales) can be implemented to represent the set of planning occurrences of a given problem. On the other hand, a controlled model dimension can also be obtained through the consideration of aggregated time instances where the replication of nonaggregated events is considered. In this paper, a replication procedure is adopted. This considers a detailed description of events, resources, and materials within an aggregated time scale. To balance between the time scale used at the planning problem and the time scale associated with general occurrences, the replication of events (tasks or flows) is allowed while observing the time and the operational requirements of each subset of the time domain (i.e., aggregated time unit, intervals, or slots). As opposed to the commonly used aggregation procedures, in this approach no clustering of events is used and therefore no statement about task sequence, operational preconditions, or aggregate capacities is necessary to represent feasible occurrences. Instead, different sequences of events can occur at each time interval (or slot) of the planning horizon. These occurrences depend on the availability of the necessary materials and resources while observing the operational requirements defined for each planning period. The replication procedure implemented is applied to tasks, flows, and depend/influence the materials and resources availability. 3.2.1. General Processing Operations—Replication of Tasks. The replication of tasks involves the following steps: (i) replication of the same processing task-allocation of a set of runs (i.e., batches) of the same processing tasks and (ii) assembly of processing tasks-occurrence of a linear combination of different task events allocated to the same processing resource; 3.2.2. General Transport Operations—Replication of Flows. The replication of flows is defined based on the compatibility of the associated transportation operations. Thus, a set of flow families corresponding to different while compatible materials flows is generated. The transportation compatibility is then stated for each pair of flow families, f and f 0 , by considering the

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suitability of the materials to be transported between a common source and sink sites. The following considerations are considered: (1) The linear combination of compatible transportation flows allocated to suitable transportation resources. The allocation of different while compatible flows is stated for both, transportation flows belonging to the same or to different while compatible flow families. (2) A sharing mode of operation is allowed for the assignment of each flow family to a given transportation resource (each transport equipment may be shared by the whole set of flows defined within a family, during a specific time domain). Based on the former replication procedures a new concept is introduced, the residence time of events at each time interval or slot. This represents the amount of time used by each event to perform a given number of e event replicas, using resource r, during a time interval or slot. At the discrete formulation the Ne is an a priori defined value (i.e., Ne = ΔP/^pte for a generic event e) since the duration of discrete time intervals and the execution time of single events (i.e., processing time of tasks i (^pti) and transportation time of flows (^t tl)) are formulation parameters. Although, at the continuous time approach the duration of each time slot is unknown and no a priori upper bound can be defined for the number of event replications (i.e., NeR e (TR - TR-1)/^pte and therefore the maximum number of e events' replicas (NeR) is a function of the slot duration (TR - TR-1) as will be discussed later on at the model formulation constraints (see section 5). Furthermore, the residence time of events (θerî or θerR) are used to ensure the fulfilment of the time requirements (time intervals and slot duration) while guaranteeing the feasibility of the events assignment resulting from its possible replication. Also, some feasibility requirements have to be guaranteed for both time formulations as will be discussed at the model formulation section (section 5.1). 3.2.3. General Material States and Resources. The replication of events previously considered can only occur if the requirements on the associated material states are verified at the beginning of each time interval or slot. The same availability conditions are applied to the resources used to perform each one of those events. Also, the products obtained through the processing tasks or transportation flows will be available only at the end of the time interval or slot. Therefore occurrences of any of these events’ replication imply a nonavailability of the material and the resources involved, during the time interval or slot. Finally, events are allowed to last longer than their execution times if no demand exists on the associated material and/or resource. As a final remark, it is important to notice that the implemented replication procedure can be used for different details on the description of events (tasks or macrotasks and flows or macroflows) within an aggregate time domain (intervals or slots). In this paper and in order to emphasize the scope of the planning formulations the events representation stands at the simplest detailed description: tasks and flows (replication procedure). Although the proposed representation (i.e., replication procedure) can also be extended to other events description, namely, aggregated events, and the proposed formulations can be easily adapted for other aggregate events description. 5009



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Chart 1. Time Horizon

4. PROBLEM DEFINITION Based on the SC characteristics previously described, the planning problem can be stated as: Given Time Horizon

Discrete Time: Discrete time intervals ΔP (e.g. 1 week)

Hp (e.g. 3, 6 months)

Continuous Time: Number of time slots R (e.g.

Resources

number of feasible subsets of the time horizon) Capacity of processing units and transportation

i∈I; l∈L; f ∈F

Resources units and facilities

j∈J; v∈V

levels; handling conditions, feasibility of recovery, etc.);

Material states

s∈S

Amounts / Due dates - Deliveries and Receipts (e.g.

The constraints on both mathematical formulations are organized as follows: (1) time horizon, (2) variable bounds (deliveries, receipts, and existences), (3) modeling events (tasks and flows), and (4) material balances. 5.1. Constraints for the Discrete and Continuous Formulations. 5.1.1. Time Horizon. The time representations used to describe and model the planning horizon (time domain) are characterized in Chart 1.

fleet of vehicles, etc.); Operational Requirements on tasks and flows (e.g. reaction, mixing, packing, inventory, transportation, etc.); Material States

Markets

Bounds, Requirements and Recovery (e.g. inventory

receipt of 5 containers of additive drug A at plant I1, during the 3rd week; deliver of 5000 of serum injection solution IS to Hospital H, at the 4th week, etc.); Determine Supply chain global planning program

Formulations’ Indices Events tasks and flows

structures (e.g. reactor, mixer, packing line, tank, Events

In this paper, the proposed planning formulations, continuous and discrete formulations, consider the simplest representation of events, tasks, and flows, within an aggregated time scale where the replication of events is allowed. Both decision models account for the multilocation and multilevel nature of the SC operation. The following indices are adopted:

Assignment of events to the suitable resources at each time subset (interval or slot) of the planning horizon Storage profiles, production and recovery amounts (e.g. Plant I1 produces 1000 bottles of serum solution IS1, and 2000 bottles of IS2, . . . , at the 2nd week; distribution center WH1, delivers 1000 bottles of IS2 to consumer C2, . . . , and 250 data overdue bottles to plant I1, etc);

In Order to Achieve the Supply Chain Optimal Global Planning Profit

5. MATHEMATICAL FORMULATIONS The supply chain operation is characterized, as previously discussed, by a geographically disperse network of operations, resources, and market places. The link between the industrial problem and its formal description is carried out through the development of an adequate representation model. This is based on a set of events, resources, and material states that represent the SC operational problem.

5.1.2. Variables and Bounds Time. The time representation has different requirements at each modeling formulation. In the discrete time model the time conditions are intrinsically considered by the definition ^ ^t ¼ Δ ^ ¼ 1, with a fixed duration of time intervals, ΔPp f Δ and corresponding to the uniform split of the planning horizon. The continuous time counterpart involves the definition of a set of time slots of unknown duration, and each slot dimension has to be optimized simultaneously with the planning events. Thus, some time bounds have to be considered exclusively for the continuous time formulation in order to consistently represent the time conditions of the planning problem. The following constraints represent the time bounds arising from the consideration of periodical time breaks (e.g., weekend stops, periodical supplies, or deliveries) within the planning horizon. 5010



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Chart 2. Receipts

Continuous Time Formulation—Periodical Time Bounds Time slot dimension (UB) TR - TR

- 1 eT

"R∈A

P

Period upper bound (UB) X ^ P NRP TR0 - TR0 - 1 eT R0 eR

-1

"R∈A

ðaÞ

ðbÞ

Period upper bound (UB) P TR gðNRP - 1Þ^t þ T0

"R∈A

ðcÞ

with 0eNRP - NRP

- 1 e1

"R∈A

ðdÞ

The maximal duration of each time slot is defined in constraints a according to the operational time breaks con^ P, sidered for the planning problem. In there, the parameter T represents the duration (elapsed time between successive ^P = time instances) of any periodical break considered (e.g., T W ^P R P D ^ ^ ^ ^ T , T = T , or T = T , respectively, for weekly stops, receipt, and delivery time tables). These constraints ensure the fulfilment of the dimensional requirements defined for slot intervals but does not account for its replication (i.e., periodical occurrence) within the planning horizon. This is considered at the period bounds defined by constraints b and c. Constraints b define the upper bound conditions ensuring that the duration of all the time slots enclosed within the considered periods (i.e., left side summation) will not exceed its duration

^P = T ^W (i.e., right side term). The replication of periods (e.g., for T W W W P ^ , 2T ^ , 3T ^ , ...) is defined by NR-1 integer it results T variables (with NPR-1 = 1, ..., NP, R ∈ A) that represent the order of the period (i.e., number of periods deduced by one) occurred until time slot R - 1. On the other hand, constraints c establish the lower bound conditions representing the minimal time requirements to move between successive time slots. Finally, constraints d ensure the unitary increment of the number of periods, between two consecutive slots (i.e., right side) while also avoiding any backward values (i.e., left side). The jointly consideration of constraints b and c avoid the occurrence of events crossing any defined period bounds (Figure 6a). To analyze the feasibility of events to crossover the period bounds, lets consider an illustrative example where we take R = 1 with NP0 = 1 ∧ T0 = 0. In this case, constraints b result into ^ P S T1 - T0 e T ^ P S T1 e T ^ P. Also, from ΣR0 e1TR0 - TR0 -1 e T inequalities c, different time upper bounds can be obtained, depending on NP1 value. If NP1 = 1 we get T1 g 0, meaning that a time point is generated within the period bounds. Although, if an unitary increment is optimally defined for NP1 (i.e., NP1 = 2) the ^ P and c inequalities result into slot lower bound is made equal to T P ^ T1 g T . Thus the combined result of b and c constraints is T1 = ^ P. This happens each time the number of periods is incremenT ted, meaning that no time slot can be extended over the period bounds. As a consequence the assignment of events has to fulfill that precise time bound and accordingly no event is allowed to cross that period limit (Figure 6a). Here, it is important to remember that events, independently of the time representation used, are balanced exclusively at time boundaries (intervals or slots) and not between them. On the other hand, if we want to allow the occurrence of events to cross the period time bounds (Figure 6b) we should 5011



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Chart 3. Deliveries

allow time slots to overcome those time limits. This is presented next and modeled through constraints e and f. X ^ P ðNRP - 1 þ YRP Þ TR0 - TR0 - 1 eT "R∈Α ðeÞ R0 eR

with YRP eNRP - NRP

-1

"R∈Α

ðf Þ

The YPR variables represent time controllers that allow the increment of the number of periods without enforcing the change of the corresponding time upper bound (constraints e). Using the former example to illustrate these new conditions, it can be noticed that, if YPR = 1 constraints e can result into ^ P S T1 - T0 e 2T ^ P S T1 e 2T ^ P while ΣR0 e1TR0 - TR0 -1 e 2T P P ^ , and thus T ^ e T1 e getting the same eq c inequalities, T1 g T ^ P. These allow the time slot to be extended further than the 2T period bound, and the assignment of events is allowed to cross that period bounds. Otherwise, if a nil value is optimally defined for YPR, this means that no SC event requires the crossing of the period bounds. These time conditions are detailed in what follows during the development of model constraints. Constraints b-d are considered for receipt and delivering conditions. Constraint b could be replaced by constraints e-f while keeping inequalities c-d although this is not considered in the developed formulation so as to ensure the consistency between both mathematical representations.

As a final remark, and for both mathematical formulations, it is important to note that the time bounds corresponding to periodical breaks (e.g., weekly stops, receipt, and delivery time tables) have ^P f T ^ 0 P;T ^R f T ^ 0 R or T ^D f T ^ 0 D). These similar notations, namely T are defined in real time units at the continuous-time approach and ^ at the ^ P, T ^ D, T ^ R gΔ) in discrete units (i.e., integer multiple of , T discrete formulation. Receipts. The optimal receipts and deliveries obtained at the planning level for each material state, s ∈ SR, at each subset of the time domain (interval or slot), result from the operational conditions presented in Chart 2. The inequalities in Chart 2 describe a common set of operational conditions modeled according to the time representations considered (constraints 1d-4d and 1c-5c, respectively, for the discrete and the continuous representations). Two major types of material supplies are considered: optional and contracted supplies. The former represent the possibility of having material receipts performed within allowable amounts (constraints 1d and 1c) and are defined as upper bound conditions to the material states receipts, at each planning interval or time slot. On the other hand, contracted supplies represent predefined planning conditions defined for the material states receipts. Our approach considers two types of contract options based on regular time tables (e.g., weekly, monthly, etc.). These are due date receipts (e.g., weekly supply of additive A) and material ^ (e.g., any ^ R gΔ supplies performed during a predefined period T time intervals (or slots) fully contained within a receipt period). 5012



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Chart 4. Material States Inventory

In both cases, contracted receipts consider minimal and maximal supply amounts that represent, respectively, the amount of each material state contracted (lower bound, LB) and any feasible extent of that amount (upper bound, UB). This is modeled through constraints 4d and 5c for the due date receipts and 2d-3d and 2c-4c for the remaining cases. When a regular supply time table is considered the material receipts can be performed during the time intervals (or slots) defined within the period duration (right-hand side of constraints 2d-3d and 2c-3c), and therefore some additional flexibility is allowed for the receipt distributions. Although, if a specific receiving time ^ 0 R þ 1 or TR = (i.e., receipt due date) has to be considered, ^t = T R ^ to some R ∈ Α, the material receipts should be performed T as a single order, at those time intervals or slots (constraints 4d and 5c). Deliveries. For the delivery events a similar modeling approach is deducted, which is shown in Chart 3. The conditions in Chart 3 describe the bounds on the material deliveries for both planning formulations, constraints 5d-8d and 6c-10c. As it was referred for the material receipts, the material deliveries can also be optional or result from a prior contract. Optional deliveries (constraints 5d and 6c) represent the possibility of SC partners to deal with nonprogrammed supplies as well as with other markets than its usual costumers. These are defined in terms of the maximal amounts (upper bound (UB)) allowed for each deliverable material, s ∈ SD, during each planning interval or time slot. Also, conditions are imposed on the global amounts delivered during a regular period (constraints 6d-7d and 7c-9c). For contracted deliveries having a defined due date, prior planning conditions are defined to represent the minimum amounts of each material state contracted (lower bound (LB), for that period, constraints 8d and 10c, as well as some maximal allowable amounts (upper bounds at inequalities 8d and 10c). Material States Inventory. On the material states inventory the constraining issue is the associated resources’ availability. This is limited by the number of existent resources and by its capacity. In these the constraints in Chart 4 are considered. The bounds on the material states amounts are defined through constraints 9d-11d and 11c-13c respectively, for the discrete and for the continuous formulations. Note, that a material state is defined as a given amount of material associated with a certain location (e.g., medicine drug A in warehouse space j Þ) is considered W). For each material state a usage factor (^ j_ ð^ on the upper and lower bounds of the material stored (constraints 10d-11d and 12c-13c) to account for the storage resource available capacity.

5.1.3. Modeling Events. Most of the planning approaches disregard operational aspects such as processing and transport operations. These are usually considered in terms of a global capacity on SC partners and therefore some operational aspects related to the coordination between available resources and the associated operations are dismissed. The aim of the proposed formulations is to extend the model in order to integrate partner specificities and options within a global decision procedure. A centralized optimization procedure devoted to the SC operational planning is developed. This is translated in terms of both proposed representation approaches. The modeling events considered characterize all suitable SC occurrences such as the processing events, defining any feasible process operation (e.g., reaction, mixing, packing, etc.), and the transportation events, representing the transfer of materials among SC partners. Processing Events. For processing events (general transforming and inventory tasks) the planning constraints in Chart 5 are integrated into the mathematical formulations. At the planning formulation the assignment of the task events (i.e., batch and semicontinuous tasks) is defined taking into account the concepts previously presented for the tasks replication and residence times (see section 3). The constraints of both formulations are organized in two major groups: inequalities 12d-15d and 14c-18c representing the operational conditions defined for batch tasks and 16d-19d and 19c-22c characterizing semibatch operations. In each one of these groups sets of modeling constraints were developed to account for the capacity requirements of the involved resources while allocated to a single task or to a set of suitable task events during each time interval or slot duration (e.g., constraints 12d-13d and 14c-15c for batch tasks and 16d-17d and 19c-20c for semibatch operations). Also, due to events replication the remaining conditions on batch and semibatch tasks are related with the replica of those tasks and with time feasibility conditions. The major difference between the presented formulations constraints’ is related to the feasibility of task replications. As it was previously reported, at the discrete formulation, the maximal number of task replications is a priori defined. This is because the duration of discrete time intervals and the tasks processing times are formulation parameters and thus the upper bound on task replicas depends exclusively on the task assignment (constraints 14d). At the continuous time approach, the duration of each time slot is unknown, and accordingly the maximal number of task replications (constraints 16c) depends on the task residence time, which is unknown, and it has to be optimized simultaneously with the time slots duration. Thus, at the continuous time formulation the upper bound on the residence time of batch and semibatch tasks is the 5013



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Chart 5. Processing Events

planning horizon, constraints 17c and 21c, respectively. This means that a task i is generically allowed to fully occupy a suitable resource during the whole planning horizon. Although, since the same time conditions should be met by both formulations, when periodical breaks are considered these upper bounds results constrained and equal to the period duration. The occurrence of such an event depends exclusively on the defined operational requirements and on the optimization criteria. Note that the time feasibility of the assignments performed during each time interval or slot is ensured by the summation of the residence times of the tasks (batch and semibatch) occurring during that subset of the time domain (constraints 15d and 17d; 18c and 20c). Concerning resources capacity, the feasibility of the tasks assignments is accounted for by the global amounts processed by each one of the tasks occurring during the considered time interval or slot (constraints 15d and 19d; 17c-18c and 22c). Transportation Events. Transportation events consider all the material transfers among SC partners and from them to customer regions. These, are modeled by a set of transportation flows or simply flows at both planning formulations. As it has been referred, the materials transportation can be performed using different operational details. In this paper and in order to keep consistency with the remaining events representation the planning formulations use a detailed flow description over an aggregated time representation.

Therefore, the constraints in Chart 6 are defined to represent the requirements on transportation events, at both formulations. The requirements on the transportation of materials have a homologous description at both mathematical formulations. The amount of materials transported at each suitable transportation resource and time period (time interval or slot), as a result of the assignment of a single flow or its replications, is limited by the resource capacity (constraints 20d and 23c). Also, the number of replications of each flow during a time period depends on the period duration (fixed for the discrete approach) and on its transportation time (constraints 21d and 24c). For the continuous approach, the planning horizon is the upper bound limit for the residence time of flows, as previously stated for processing tasks, if no constraining condition is added up (see 25c). The remaining constraints translate the feasibility conditions for the occurrence of compatible flows sharing a suitable resource. The capacity bounds are ensured by constraints 22d and 26c, while the possibility of family flows’ replications is accounted at inequality constraints 23d-24d and 27c-30c. Finally, the time feasibility of the assignments performed during each planning period, to each transportation resource, is defined in constraints 25d and 31c. The whole amount of material transported by a given flow during a time interval or slot is defined by inequality constraints 26d and 32c. Modeling of Material Recoveries. The modeling approach developed considers the integration of materials flows resulting 5014



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Chart 6. Transportation Events

from nonconformed materials returned into supply chain sites. These are usually supplied by customer to supply chain sites (usually to distribution sites closer to the final customers’ location) due to different nonconformity reasons. Thus, to account for the recovery of such a large diversity of materials different processes are required. Independently of the proprieties of the returned materials, these are considered as material entrances into different SC sites ^^s^t (R ^ sR) at both SC and are modeled as external receipts, R formulations. Also, after the returned materials enter any SC site, the recovery process starts with their transportation to a partner where this can be performed. In the mathematical formulation, the flow of returned materials is considered through the definition of a set of transportation flows, usually with the origin at a distribution site and the ending at plant facilities, and defined over a suitable transportation structure. These material flows are added up to the set of feasible transportation flows and their operability conditions are accounted for through the constraints previously defined. At the supply chain sites, the materials are separated according to their nonconformity and with respect to their feasible final purpose (e.g., recycling, remanufacturing, or disposal). These procedures are modeled by a set of processing tasks, ^ pij^t (Q ^ pijR), defined on the materials separation and on the Q feasible remanufacturing operations. The output materials of each one of these tasks are stored at an inventory facility, defined by ^Ss^t (^S^sR) variables, or either transported to distribution ^ Λl^t (Q ^ ΛlR), or delivered to the other external markets, sites, Q ^ s^t (D ^^sR). D

Material Balances. For each storable material state, s, a material state equation is defined, for each time interval, ^t , or time slot, R. These equality constraints relate the amount of each material state, ^Ss^t (^SsR), with the material existences in the prior time interval or slot, ^Ss^t-1 (^SsR-1), and with the amounts being ^ pijR), transported into and ^ pij^t (Q produced and consumed, Q ^ ΛlR), as well as with the amounts ^ Λl^t (Q from the state, Q ^ sR) and receipt, R ^ s^t (R ^ sR), into/from the state ^ s^t (D delivered, D at the considered time interval or slot. As it can be observed in the Figure 7, the bill of materials is defined exclusively at the interval or slot time bounds, while allowing events replication within those time periods. Discrete S^^Ss^t ¼ S^^Ss^t - 1 þ

X

^ Λl^t Q

X

X

^ pij^t ϑsi Q

^ s^t - D ^D ^ s^t "s∈S0 , þR

l∈SIsout

^ Λl^t þ Q

X

X

X

"^t ¼ 1, :::, H þ 1

X

j∈J i∈JIj ∧i∈SIsout

ð27dÞ

^ pij^t ϑsi eS^^Ss^t - 1 Q

"s∈S0 , "^t ¼ 1, :::, H þ 1 5015

^ pij^t ϑsj Q


j∈J i∈JIj ∧i∈SIsin

X

^ Λl^t Q

l∈SLout s

l∈SLin s

þ

X

ð28dÞ



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differentiated by the notation details previously considered. Thus, to avoid repetitiveness the economical balance is performed for the discrete formulation and accordingly a single number is place at each term. In each economical term the index ^t has to be replaced by R for the continuous time formulation. 5.2.1. Economical Incomes. The economical incomes consider the revenues resulting from the material states produced and kept in storage at each time, material assets, E1 and from the materials sold to costumers, material states’ delivers, relations E2.

Figure 7. Balance of materials existences.

Continuous ^S^SsR ¼ ^S^SsR þ

X

-1

þ

X

^ ΛlR Q

l∈SLin s

X

^ pijR ϑsi Q

X

X

^ ΛlR Q

l∈SLout s

X

^ pijR ϑsi Q


j∈J i∈JIj ∧i∈SIsin

^D ^ sR þ R ^ sR -D X l∈SIsout

^ ΛlR þ Q

X

X

^ pijR ϑsi e^S^SsR Q

ð33cÞ -1

"s∈S0 , "R∈Α


The material assets, E1, as well as the deliveries supported by the customers (no cost incurred) are balanced at the ex-work price, pws, first summation (1) on terms E2. Although, when the deliveries of materials are performed by the sourcing SC partner, the income term is balanced at the market price, pms, second summation (2) on terms E2. 5.2.2. Operational Costs. The cost issues considered at the objective functions comprise the costs resulting from (i) the external material’s receipts, (ii) the supply of raw and auxiliary materials, and by (iii) the occurrence of SC events as storage, processing, and transportations.

ð34cÞ The equality constraints 27d and 33c establish the balance of materials at each mathematical formulation. Due to the possibility of having replication of events some additional constraints are considered in order to guarantee the feasibility of the corresponding assignments. These are represented by constraints 28d and 34c, which ensure that the amount of material consumed by a single task or any feasible assembly of tasks (e.g., a batch or a set of batches), added with to the amount of that material transported (i.e., leaving the state) by a single flow or by a set of replicas, can never exceed the amount of material remaining at the former time interval or slot. This means that the assignment of events is done, at each time interval or slot, based on the materials availability at the prior planning period. 5.2. Objective Functions. The objective function considered at both developed formulations 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. with objective function, OF, ¼ incomes - costs

max z : z

ð29d - 35cÞ

incomes ¼ material assets þ deliveries costs ¼ materials þ events þ receipts þ storage þ transportation þ processing The economical sources considered at the objective function are balanced at both SC formulations through similar terms

The materials’ supply is balanced at the source price, pws, for the raw materials E3-1 and with an incremented price for other materials fulfilled to SC sites. The remaining cost issues are related to the occurrence of SC events where a fixed and a variable cost term are considered. For processing events, the fixed costs depend on the resources allocation to any suitable event, in accordance to the number of event replicas performed (terms E5-1), while the variable costs depend on the assigned amounts, terms E5-2. Also, for semibatch processing tasks only fixed costs are stated. These are determined based on its residence time since no fixed processing time is a priori defined. 5016



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Figure 8. Supply chain considered in the case-study.

On the other hand, for the transportation resources a fixed cost representing the maintenance of transportation structures (i.e., first summation on E4) is considered which is added up to a set of variable costs defined by the resources allocation to any suitable flow, l, or set of compatible flows f (i.e., flow family). Finally, the storage cost terms consider fixed expenses resulting from owned capacities, as equipment maintenance (vessels, tanks, warehouse spaces, etc.), and variable costs, depending on the amount of each material stored, Sst, and on its unitary storage cost, SCs. In conclusion, the objective functions 29d and 35c, respectively, to the discrete and continuous formulations, together with the constraints, 1d to 28d and 1c to 34c, define respectively the MILP formulations developed for the SC planning problems.

6. INDUSTRIAL EXAMPLE We will now look into a particular case of supply chains, the pharmaceutical chains. These, independently of the market dimension and complexity, are characterized by a set of industrial positions (e.g., production cluster, plants I1, I2, and I3, Figure 8) mainly focused on multipurpose batch plants that produce a wide range of medicine drugs. This case is solved using the two formulations presented that were implemented using the GAMS language and solved with a Branch and Bound (B&B) procedure (CPLEX). 6.1. Supply Chain Description. The industrial case follows a former study developed for a pharmaceutical supply chain, SC, presented by Amaro and Barbosa-Povoa,18 although different supply chain operational conditions are here considered. The SC in study involves the production and distribution of different injection drugs, tablets, and oral suspensions for different Portuguese and international markets. The production cluster comprises three medicine manufacturers (I1 and I2, Portuguese partners, and I3 a Spanish partner). Plants I1 and

I3 produce four major intermediate medicines, IP1 to IP4, while I2 produces only IP3 and IP4. On the basis of the market and legislation requirements three types of medicines customizations are defined: (i) PF, exclusively for the Portuguese and African markets; (ii) SF, exclusively for Spanish market and associated customers; and (iii) EF, for a set of other European (EEC) market positions. The customization processes can be performed at any industrial plant directly from the intermediate materials, produced at that production site, or from any intermediate material formulated at any other production partner. The raw medicines and general raw-materials (e.g., packing materials, glass bottles) are received at the plants’ warehouses and their transport is not a current supply chain concern. On the other hand, the logistic distribution of materials among partners is performed by three transportation fleets controlled by the supply chain in analysis. The first one is planned for long-range travels or higher capacity charges, π = 1, while the other two, π = 2 and π = 3, are used for lower charge capacities. Reserve flows accounting for products with nonconformities are also considered. The closed loop supply chain operation is evaluated by comparing the supply chain economical performance for a given aggregated planning period where three independent operational scenarios are studied. A first scenario represents a disposal scenario where all the nonconformed products are sent to burning centers (while removed from the market places). The latter two scenarios consider a recovery planning approach where the medicines are grouped based on the nonconformity. Nonrecoverable medicines are sent to burning centers while recyclable or remanufacture medicines can be recovered at the I1 plant. Scenarios II and III consider, respectively, an unconstrained medicine recovery option and a recovery practice involving minimal requirements for all recoverable medicines (III). 5017



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Table 2. Optimal Results and Statistics for CTF at the Disposal Scenario, Sc I

Figure 9. Discrete and continuous time grids representations.

6.2. Optimal Planning Results. Based on the above char-

acteristics the SC planning is performed for a planning period of 3 months with the objective of evaluating the best recovery portfolio scenario. An aggregate time description is used in both formulations. For the discrete time formulation (DTF) a uniform time grid, defined on a weekly basis, is used to represent the planning horizon, Figure 9. At the continuous time formulation (CTF) these weekly operational breaks (weekends) also represent the SC operability conditions (partners working schedules). Therefore, a fixed set of common time points is generated for both formulations but in the continuous one a detailed time description (refined grid) is used. This accounts for the generation of additional time evaluations enclosed within each weekly duration (i.e., periodical stops) fact that is not allowed at the discrete time model (see sections 3.1 and 5.1 and Figure 9). 6.3. Implementation of the Time Representations. To better understand the optimal results obtained with both mathematical formulations, some considerations on the time representation have to be made. These are related to previous tests performed over the proposed models with the objective of testing its consistency. First, to ensure a coherent representation of the planning problem with both mathematical formulations a test on both models was done using a reference scenario, the disposal scenario. The same number of time intervals or slots (weekly periodical stops) was used in both formulations and the obtained results showed a very small difference (0.20%) between the values of the objective functions of both formulations. This difference is explained by the time representation for the semicontinuous tasks at each model formulation (integer versus continuous durations) and to their subsequent economical evaluation. On the basis of these results the consistency of the models representation is confirmed. After this consistency analysis a set of computational tests, exploring time evaluations within each weekly periodical stop, was performed for the continuous time approach. These tests aim to identify the minimal requirements on the number of time slots R for the continuous formulation. Different number of time slots were analyzed (between 13 and 20). Table 2 illustrates the results of these tests for the disposal scenario. The computational statistics and the economical performances obtained at the optimal solution for the disposal scenario (see Table 2), show that R = 14 is the best value since it presents the highest optimal value with the lowest relative gap (RG = 0.70%). Some equivalent tests were done for the remaining operational scenarios and the “optimal” values obtained were, respectively, 14 and 15 for scenarios II and III. Thus, the implementation of a number of time points larger than that required to generate those time slots will increase the model dimension reducing the computational performances without improving the optimal planning solution. As it can be observed in Table 2, the CTF presents a better optimal value, obtained at the expense of larger CPU times (almost 3 times of the discrete value) and with a relative gap also larger than the one obtained for DTF.

best integer

absolute gap

relative gap

optimal values

OF (m.u.)

(m.u.)

(abs G, m.u.)

(RG e 1%)

DTF 12 Δ CTF slots

18 690 558

18 810 416

119 858

=0.58%

13

18 877 600

19 013 130

135 530

=0.72%

14

18 883 129

19 015 652

132 523

=0.70%

15

18 811 714

18 995 925

184 210

=0.98%

constraints

variables

integer

CPUs

30 801

17 355

6 838

=430

13 14

43 641 46 951

21 435 22 993

6 997 7 535

=583 =1159

15

50 261

24 551

8 073

=1250

statistics DTF 12 Δ CTF slots

Table 3. Optimal Results and Model Statistics for the CTF (14 slots) and the DTF (Δ = 12 and 24) at the Disposal Scenario periodical bound (amounts) weekly maximal fourthly minimal and maximal

delivering and receipts

DTF 12 Δ

formulation

DTF 24 Δ

CTF 14 slots

Model Statistics equations

30801

54741

continuous var.

17355

33219

21474

integer var.

6838

13666

7036

abs. G (RG e 1%) CPUs* =

43680

119858

112574

135530

(=0.58%)

(= 0.61%)

(=0.70%)

430

646

1159

Optimal Values (m.u.)

*

objective function

18690558

18517836

18883129

best integer

18810416

18630410

19015652

GAMS/CPLEX (v.10), Intel Core Duo T7300 2 GHz.

Since the discrete model performs better than the CTF in terms of the computational performance it is interesting to check how closely the optimal values of the discrete model could be to the ones obtained through the CTF. To check out this, and at the same time to verify the “robustness” of the formulation, the discrete model was executed also for 24 time intervals (i.e., each week is now described by two time points), while keeping equivalent operational conditions. The results obtained, for the disposal scenario, are present in Table 3. The DTF with Δ = 24 resulted, as expected, into a largest formulation but even though it gave the optimal solution for a smaller CPU time than the one taken by the CTF. When analyzing the objective function values the results obtained, are a bit surprising, since it would be expected that using a small value in the time discretization would result in a better objective function value, which is not the case (18 690 558 for DTF Δ=12 and 18 517 836 for DTF with Δ = 24). However, it is important to note that the results are influenced not only by the 5018



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number of time intervals but also by its duration and distribution within the planning horizon. In fact, and for this case, when the time intervals were broken twice (Δ = 24) some task replications become infeasible (e.g., if task 1 spent 8 h, the model can optimally assign 5 replications within a 40 h week, while it can optimally assign only 4 replications for a set of two time intervals of 20 h each) and therefore no objective function solution improvement was obtained. Moreover, at the CTF the slots generated with the period bound are integer multiples of the tasks assigned. This explains why the CTF provides better results. Still on the formulation issues and in order to verify the performance of both formulations other tests were made. The continuous time model was solved for a relative gap, RG, less or equal to 0.61% (the same of the DTF, see Table 3) and the discrete time counterpart (DTF) with 12 time intervals was solved for 1159 CPU seconds (CTF CPU time, see Table 3). For the CTF, the relative gap, RG, obtained was 0.59% after 1300 CPUs while the DTF formulation showed a RG of 0.50% after the defined 1159 CPU time. The optimal solutions get a bit closer to each other but there still exists a gap around 1% between the two optimal solutions (the optimal value for DTF is 18 709 626 m.u.). 6.4. Operability and Economical Analysis. After analyzing the formulation issues, the identified formulation conditions were used to run the operational scenarios above identified (Sc I, II, and III). The economical results obtained using both mathematical formulations are present in Table 4. The analysis of these results will be made considering two main aspects: (i) the differences between CTF and DTF optimal results, at each operational scenario; and (ii) the differences between operational scenarios, at each formulation. First, it should be noticed that the results of both formulations, CTF and DTF, follow the same global behavior, with the CTF presenting optimal values a bit higher than the DTF, for all the studied scenarios (Table 4). When the results are analyzed globally, it can be stated that the recovery of nonconformed materials is always the profitable operational policy to follow (Table 4). For a recovery situation when a nonconstrained recovery option is accounted (Sc II), a higher planning profit is obtained (19 031 109 and 18 872 028 monetary units, m.u., respectively, for the CTF and the DTF). Table 4. SC Profit Obtained for the Studied Formulations and Planning Scenario formulation

Sc I (m.u.)

Sc II (m.u.)

Sc III (m.u.)

CTF OF

18 883 129

19 031 109

RG (CPUs)

0.70% (1159)

0.74% (837)

18 911 935 0.81% (1244)

DTF OF

18 690 558

18 872 028

18 813 876

RG (CPUs)

0.61% (430)

0.55% (131)

0.76% (409)

ΔCTF-DTF

192 571

159 081

98 059

CTF

ΔII-I = 147980

ΔII-III = 28806

ΔIII-I = 119174

DTF

ΔII-I = 181470

ΔII-III = 58152

ΔIII-I = 123318

Δ Scenarios

The profit values observed for the constrained recovery scenario (Sc III) are smaller than the latter ones (ΔII-III = 28806 m.u. and ΔII-III = 58152 m.u., respectively, for the continuous and the discrete formulations) but higher than the profit values obtained for the disposal scenario (Sc I). The observed results allow us to conclude that in this case the recovery must be the chosen industrial practice (Sc II is profitable than Sc III). This should account for the full recovery of the profitable nonconformed products while satisfying the regulatory legislation of the remaining materials (selective recovery). When the results of each one of the formulations developed are analyzed in detail, it can be seen that the continuous formulation presents the better optimal planning solutions for the set of studied scenarios. The differences observed are not C however very significant ((OFCSci - OFD Sci)/OFSci e 1%) when comparing the optimal value of the objective functions achieved. The continuous formulation performs economically a bit better due to the increment of the number of time points generated to achieve the optimal solution. At these time points the existences of materials are actualized and a different feasible set of tasks can result for the next, and further, time slots. 6.5. Detail Analysis on the Planning Results Obtained. To explain the major economical and operational differences between the formulations the disposal scenario was chosen to be analyzed in more detail. In there, the contribution of each economical aspect is detailed as it is presented in Table 5. The main economical differences are explained by the existing balance between the material assets (exist1 and exist2) and the material receipts performed during the planning horizon. These have opposite contributions to the objective values and present a small difference in terms of absolute value. Since the DTF has higher economical receipts than the CTF, the material assets due to the period are also larger than the ones of the CTF. Moreover, these material assets should result from receipts performed closer to the end of the planning horizon since the storage costs at the DTF are smaller than the ones observed at the CTF. Thus, different planning decisions are considered at both formulations. To better understand these planning decisions, the amounts associated with each one of these economical terms were observed and are presented in Table 6. From this table it can be seen that the quantities associated with the material assets and receipts do not explain the economic values obtained. These are however explained by the existence of transformation of the receipt materials as it can be verified through the values of the processing operations, form.cv and custom., Table 6 (note that there is not a direct conversation of materials from one-to-one but some conversation factors exist, 1 unit of active medicine drug is transformed into 1000 units of soft tablet medicine). On the processing operations performed, at both formulations, it can be seen that the DTF results show the occurrence of a small number of operations (no. batch) with higher amounts and having a lower cost, when compared to the CTF formulation. Thus, globally the formulations result in similar planning profit

Table 5. Optimal Results for the Economical Terms of the Objective Function of CTF and DTF, at the Disposal Scenario exist1

exist2

deliver

recp.1

recp.2

store

transp

fm. cv

no. batch

custom.

Ec.CTF

25 296 800

694 500

32 257 090

22 980 230

878 500

187 314

1 742 525

5 344 037

1 635 573

6 630 015

Ec.DTF

29 149 700

688 850

32 168 540

26 852 260

878 500

180 815

1 856 178

5 213 407

1 584 307

6 698 236

-3 852 900

5 650

88 550

-3 872 030

0

6 499

-113 653

130 630

51 266

-68 221

ΔCTF-DTF

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Table 6. Optimal Amounts of Materials Involved in CTF and DTF, at the Disposal Scenario quantity

exist1

exist2

delv

recp1

recp2

store

transp

fm. cv

no. batch

custom. 4 134

A.CTF

3 031 431

69 000

2 752 345

34 114 840

88 250

10 832 830

3 325 000

5 270 833

687

A.DTF

6 335 765

68 500

2 735 655

34 279 610

88 250

10 949 250

3 493 578

5 279 189

673

4 255

-3 304 334

500

16 690

-164 770

0

-116 420

-168 578

-8 356

14

-121

ΔCTF-DTF

Table 7. Summary of Formulation Statistics for the CTF and the DTF Sc I formulations

*

Sc II

Sc III

CTF

DTF

CTF

DTF

CTF

DTF

equations

46951

30801

50253

32817

53797

32821

variables integer

22993 7535

17355 6838

24719 8053

18547 7234

26391 8628

18599 7234

R.G (e1%)

0.70%

0.61%

0.74%

0.55%

0.81%

0.76%

slots/ Δ

14

12

14

12

15

12

CPUs*

1158

430

837

131

1244

409

GAMS/CPLEX (v.10), Intel Core Duo T7300 2 GHz.

values although such values correspond to different planning decisions such as time of deliveries and occurrence of tasks as well as amounts involved, among others. 6.6. Model Statistics. Concerning the analysis of the models dimension and computational statistics the continuous formulation has a performance globally worse than the discrete time counterpart, in any of the studied scenarios (Table 7). The number of continuous and integer variables at the continuous time formulation as well as the number of constraints is larger than the corresponding ones at the discrete model. This explains the statistics observed for both formulations in what concerns the number of iterations and CPU requirements with a relative gap less than or equal to 1%. Through the solution of this case it can be seen that although the continuous formulation may lead to more profitable solutions the CPU consumption is always higher than the one obtained for the discrete formulation. Nevertheless, it is important to notice that due to the dimension and characteristics of the industrial case study the comparative analysis of both mathematical formulations is really complex.

7. CONCLUSION This paper presents two mathematical formulations that were developed and implemented to model the supply chain optimal planning problem: a discrete and a continuous-time formulation. Both formulations account for detailed supply chain characteristics such as production, storage, distribution, and recovery of products. Different recovery portfolio scenarios were analyzed as well as delivery policies. A real case study taken from a pharmaceutical industry was studied, and the results obtained help to concluded about the performance and characteristics of both developed formulations. The discrete formulation performs better than the continuous counterpart in what concerns the model statistics. It requires less iterations and less CPUs to reach the optimal solutions, although the continuous counterpart gives a marginally better optimal solution for the planning conditions studied. The improvement on the net global profit observed for the continuous time model is due to the increment of the number and location of the

evaluated points on the time domain fact that is not allowed at the discrete model since a fixed discrete time is used. The developed study brings up some interesting points that require further attention. First, it is important to apply both models to other cases so as to generalize the results obtained. Also, an interesting future development is related to the comparative analysis of the proposed continuous model and the one resulting from the integration of crossing conditions in its formulation. This represents an extension of the continuous approach developed, and some profitable solutions can be expected. This will result in larger formulations due namely to the associated increment of the model variables, and thus some additional investment has to be done to ensure robustness against an enlarged dimension.

’ NOMENCLATURE This paper presents different time representations while defining the mathematical planning formulation: the discrete and the continuous time representation. To avoid a huge list of formulation symbols a compact notation rule is considered to describe the mathematical entities used at both developed formulations (see Table 8). The implemented approaches consider a detailed events’ description within an aggregated time scale (time intervals and slots). Thus, common symbols are used to represent homologous entities involved at both planning models, the distinction between them, if any, is posted at the time index (e.g., ^t and R). Indices.

i = 1, ..., Ni processing tasks j = 1, ..., Nj processing resources l = 1, ..., Nl- transportation flows s = 1, ..., Ns- material states v = 1, ..., Nv transportation resource π = 1, ..., Nπ transportation structures ^ 0 | planning time intervals ^t = 1, ..., |T R = 1, ..., |Α| planning time slots Simple Sets (or Structural Sets). Table 8 includes simple sets (or structural sets). Set IB considers all the batch tasks that can occur at some suitable resource (unit or facility) belonging to the set of processing resources, JP. Also, for storable materials, S0 , a set of storage capacities, units JS, is defined. Composed Sets. Table 9 lists the composed sets. Continuous/Integer Variables. ^ s^t (D ^ sr) = amount of material state s delivered to external custoD mers, at the beginning of time interval ^t or slot R ^ ΛlR) = global amount of material transported by flow l, ^ Λlt(Q Q using a set of suitable transportation resources, at the beginning of time interval ^t or slot R ^ λvlR) = total amount of material transported by flow l, ^ λvl^t (Q Q assigned to resource v, at the beginning of time interval ^t or slot R, while accounting for any feasible flow replication 5020



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Integer Variables.

Table 8

NRr(ND r)

indices symbols time intervals slots

^0, R ∈ Α ^t ∈ T

equipment unit/facility

j ∈ J, j = 1, ..., Nj

task

i∈I, I = 1, ..., Ni

material states

s ∈ S, s = 1, ..., Ns structural sets

I (IB ⊆ I)

J (JP ∪ JS)

JR(J ∪ JS)

(Πo ⊆ Π)

L 0

a

(S0 ⊆ S)a

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

Table 9 details {element ∈ Set (such that): condition}; |Set| (if required)

set

{l ∈ L: flow l is defined through transportation structure π} {i ∈ I: processing task i can be performed at resource j ∈ J} {s ∈ S0 : material state s that 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: set of compitable flows}; {f: f is a flow family suitable on transportation structure π};

Fπ JIj STj out Sin i /Si out SIin s /SIs out SLin s /SLs Γπ

FLf FVπ

^ pijR) = total amount of material processed by task i, using ^ pij^t (Q Q resource j, at the beginning of time interval^t or slot R, while accounting for any feasible task replication ^ sR) = amount of material state s received from the outside ^ s^t (R R market into a supply chain site, at time interval ^t or slot R ^Ss^t (^SsR) = amount of material state s, a material in a given location, available at the beginning of time interval ^t or slot R SB (θ θBijR/θSB ^ ijt ijR) = residence time of batch/semibatch task i, done at unit j, during planning period ^t or slot R θLlvR = residence time of flow task l, done at resource v, during slot R ΘfvF^t (ΘFfvR) = residence time of flow task l, done at resource v, during period ^t or slot R Binary Variables. Y^ λvl^t Y^ λvlR ¼

8 > 1 > >
> > :0

otherwise

8 >1 > < Y^ pij^t Y^ pijR ¼ > > :0

Y^ νv^t Y^ νπR

¼

8 > 1 > < > > :0

if processing task i is assigned to equipment j, at time interval ^t or slot R otherwise

if transport resource v is occupied at time or slot R, with the transport of some material ðsÞ otherwise

= order of the supplying (delivering) period performed until time slot R (i.e., number of periods deduced by one) NijB^t (NBijR) = number of replications of tasks i, done at unit j, during planning period ^t or slot R NLlv^t (NLijR) = number of replications of flows l, done at unit v, during planning period ^t or slot R NFfv^t (NFfv^t ) = number of replications of family flows f, done at unit v, during period ^t or slot R Time and Capacity Parameters. Cj = capacity of the processing resource j (volume, mass, rate, etc.) CVv = capacity of transportation resource v (volume, mass, etc.) ^ ^ ðD _ s , D s Þ = minimal/maximal amount (rate) of material state s that can be delivered to external customers. These represent optional delivers corresponding respectively to the minimal market demand level and the maximal market capacity ^C ^C ðD _ s , D s Þ = These represent the bounds for contracted deli^ 0 D þ 1, vers and are defined for time intervals ^t = T D ^D 0D ^ ^ 2T þ 1, ..., or slots R ∈ A: TR = T ,2T , ... ^pti = completion time of processing task i. These represent the time required to fully execute a certain processing operation m = minimal operating time defined for the assignment of O i semibatch tasks i ^ ^ ðR _ s , R s Þ = minimal/maximal amount (rate) of material state s that can be sourced, by an external provider, at time interval ^t or slot R ^C ^C R , Þ ðR _s s = These represent the bounds for contracted delivers ^ 0R and are defined for time intervals ̂t = cT̂0 R þ 1, 2T ^ R,2T ^ R, ... þ 1, ..., or slots R ∈ A: TR = T ^t tl = transportation time of flow l. These represent the travel time or duration of the transport operation, l out ϑin is /ϑis = rate or proportion of state s undergoing/leaving task i, accordingly to the task recipe εl = charge coefficient of flow l φsj = proportion of storage resource, j capacity, dedicate to material state s Φij = utilization factor of the resource j, by processing event i Ψv = minimal charge coefficient of transportation resource v (structure π) Θ = [θll0 ]"l,l0 ∈L = incompatibility matrix that stores the information on flows incompatibility Ξπ = max"l∈Fπϑlπ þ 1 = maximum number of compatible flows defined in structure π Cost and Price Parameters. FCij = fixed operating cost of processing task i assigned to resource j (h/batch; h/run) FOCv = fixed transport cost of each owned resource v, (h/ resource) CCπ = fixed transport cost of a contracted transportation structure (h/ capacity) FSj = fixed storage cost of resource j ∈ JS (h/resource) OCij = size dependent operating cost of processing task i assigned to resource j (h/amount) pms = unitary retailing price or market value for material state s. Usually pms = pws (1 þ Δpws), where pws is the unitary exwork or producing price of state s and Δpws is the profit margin SCs = storage cost for material state s 5021


Industrial & Engineering Chemistry Research TCv = transport cost defined for the assignment of a contracted transport resource v γvl = cost parameters defined to evaluate in-house transport resource v

’ AUTHOR INFORMATION Corresponding Author

*Tel.: þ351214233265. Fax: þ351214 233 568. E-mail: apovoa@ ist.utl.pt.

’ REFERENCES (1) Papageorgiou, L. G. Supply chain optimization for the process industries: Advances and opportunities. Comput. Chem. Eng. 2009, 33, 1931–1938. (2) Varma, V. A.; Reklaitis, G. V.; Blau, G. E.; Pekny, J. F. Enterprise-wide modelling & optimization—An overview of emerging research challenges and opportunities. Comput. Chem. Eng. 2009, 31, 692–711. (3) Grossmann, I. E. Enterprise-wide optimization: A new frontier in process system engineering. AIChE J. 2005, 51, 1846–1857. (4) Goetschalckx, M.; Vidal, C.; Dogan, K. Modelling and design of global logistics systems: A review of integrated strategic and tactical models and design algorithm. Eur. J. Operat. Res. 2002, 143, 1–18. (5) Papageorgiou, L. G.; Georgiadis, M. C., Eds. Supply chain optimization: Parts I and II. Process System Engineering; Wiley-VCH: Weinheim, Germany, 2008. (6) Melo, M. T.; Nickel, S.; Saldanha-da-Gama, F. Facility location and supply chain management—A review. Eur. J. Operat. Res. 2009, 196, 401–412. (7) You, F. Q.; Wassick, J. M.; Grossmann, I. E. Risk management for a global supply chain planning under uncertainty: Models and algorithms. AIChE J. 2009, 55, 931–946. (8) You, F. Q.; Grossmann, I. E. Optimal design and operational planning of responsive process supply chains. In Supply Chain Optimization; Papageorgiou, L. G., Georgiadis, M. C., Eds.; Wiley-VCH: Weinheim, Germany: 2008; Part I, pp 107-134. (9) Amaro, A. C. S.; Barbosa-Povoa, A. P. F. D. The effect of uncertainty on the optimal closed-loop supply chain planning under different partnerships structure. Comput. Chem. Eng. 2009a, 33, 2144–2158. (10) Lainez, J. M.; Kopanos, G; Espuna, A; Puigjaner, L. Flexible Design-Planning of Supply Chain Networks. AIChE J. 2009, 55, 1736– 1753. (11) Sousa, R; Shah, N; Papageorgiou, L. G. Supply chain design and multilevel planning —An industrial case. Comput. Chem. Eng. 2009, 32, 2643–2663. (12) Guillen, G.; Mele, F. D.; Espuna, A.; Puigjanner, L. Adressing the design of chemical supply chains under demand uncertainty. Ind. Eng. Chem. Res. 2006, 45, 7566–7581. (13) Amaro, A. C. S.; Barbosa-Povoa, A. P. F. D. Planning and scheduling of industrial supply chains with reverse flows: A real pharmaceutical case-study. Comput. Chem. Eng. 2008, 32, 2606–2625. (14) Lakhdar, K; Savery, J; Papageorgiou, L. G. Multiobjective longterm planning of biopharmaceutical manufacturing facilities. Biothechnol. Prog. 2007, 23, 1383–1393. (15) Guillen, G.; Badell, M.; Puigjanner, L. A holistic framework for short-term supply chain management integrating production and corporate financial planning. Int. J. Prod. Econ. 2007, 106, 288–306. (16) Huang, C.; Karimi, I. A. Scheduling trans-shipment operations in maritime chemical transportation. Ind. Eng. Chem. Res. 2006, 45, 1955–1973. (17) Al-Ameri, T. A.; Shah, N; Papageorgiou, L. G. Optimization of vendor-managed inventory systems in a rolling horizon framework. Comput. Chem. Eng. 2008, 54, 1019–1047. (18) Amaro, A. C. S.; Barbosa-Povoa, A. P. D. F. A Continuous-Time Approach to Supply Chain Planning: Managing Product Portfolios

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subject to Market Uncertainty under Different Partnership Structures. Comput.-Aided Chem. Eng. 2009b, 26, 399–404. (19) Puigjaner, L; Guillen-Gosalbez, G. Towards an integrated framework for supply chain management in the batch chemical process industry. Comput. Chem. Eng. 2008, 32, 650–670. (20) Shah, N. Process Industry Supply Chains: Advances and Challenges. Comput. Chem. Eng. 2005, 29, 1225–1235. (21) Grossmann, I. E. Challenges in the new millennium: Product discovery and design, enterprise and supply chain optimization, global life cycle assessment. Comput. Chem. Eng. 2004, 29, 29–39. (22) Maloni, M. J.; Benton, W. G. Supply chain partnership: Opportunities for operations research. Eur. J. Operat. Res. 1997, 101, 419–429. (23) Shapiro, J. F. Challenges of strategic supply chain planning and modelling. Comput. Chem. Eng. 2004, 28, 855–861. (24) Puigjaner, L; Lainez, J. M.; Alvarez, C. R. Tracking the Dynamics of the Supply Chain for Enhanced Production Sustainability. Ind. Eng. Chem. Res. 2009, 48, 9556–9570. (25) Guillen-Gosalbez, G; Grossmann, I. E. Optimal Design and Planning of Sustainable Chemical Supply Chains Under Uncertainty. AIChe J. 2009, 55, 99–121. (26) French, M. L.; LaForge, R. L. Closed-loop supply chains in process industries: An empirical study of producer re-use issues. J. Operat. Manage. 2006, 24, 271–286. (27) Salema, I.; Barbosa-Povoa, A. P. F. D.; Novais, A. Q. Simultaneous design and planning of supply chains with reverse flows: a generic modelling framework. Eur. J. Operat. Res. 2010, 203, 336–349. (28) Guide, V. D. R.; Harrison, T. P.; Van Wassenhove, L. N. The challenge of closed-loop supply chains. Interfaces 2003, 33, 3–6.

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Supply Chains Planning with Reverse Flows: Optimal Alternative Time

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