Robust decision making for hybrid process supply chain systems via model predictive control

Computers and Chemical Engineering 62 (2014) 37–55

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Computers and Chemical Engineering journal homepage: www.elsevier.com/locate/compchemeng

Robust decision making for hybrid process supply chain systems via model predictive control Richard Mastragostino, Shailesh Patel, Christopher L.E. Swartz ∗ Department of Chemical Engineering, McMaster University, 1280 Main Street West, Hamilton, Ontario, Canada L8S 4L7

a r t i c l e

i n f o

Article history: Received 6 July 2012 Received in revised form 8 August 2013 Accepted 24 October 2013 Available online 20 November 2013 Keywords: Robust model predictive control Supply chain optimization Multi-objective optimization Stochastic optimization Supply chain management

a b s t r a c t Model predictive control (MPC) is a promising solution for the effective control of process supply chains. This paper presents an optimization-based decision support tool for supply chain management, by means of a robust MPC strategy. The proposed formulation: (i) captures uncertainty in model parameters and demand by stochastic programming, (ii) accommodates hybrid process systems with decisions governed by logical conditions/rulesets, and (iii) addresses multiple supply chain performance metrics including customer service and economics, within an integrated optimization framework. Two mechanisms for uncertainty propagation are presented – an open-loop approach, and an approximate closed-loop strategy. The performance of the robust MPC framework is analyzed through its application to two process supply chain case studies. The proposed approach is shown to provide a substantial reduction in the occurrence of back orders when compared to a nominal MPC implementation. © 2013 Elsevier Ltd. All rights reserved.

1. Introduction A supply chain (SC) is a network of facilities that performs the functions of raw material procurement, raw material transformation into intermediate and finished products and distribution of products to customers, traditionally characterized by a forward flow of material, and a backward flow of information in the form of demand and orders. In a chemical process supply chain (PSC), as in the petrochemical or pharmaceutical industry, manufacturing is a major component (Grossmann, 2012). Reducing working capital and operating costs, while maintaining a high level of customer service are critical for remaining competitive within highly global environments. Supply chain management (SCM) and supply chain optimization (SCO) are concerned with the efficient coordination and integration of business and operational functions, including strategic supply chain design, purchasing, production, transportation and distribution in order to bring greater net value to the customer, at minimum overall cost. Key drivers toward an increased focus on SCO in the chemical industry include increasingly global markets, reducing costs/inventories, improving responsiveness, and mitigating against uncertainty. There are significant economic incentives to be realized through improved integration between business planning and operational decision making at manufacturing facilities.

∗ Corresponding author. Tel: +1 905 525 9140; fax: +1 905 521 1350. E-mail address: [email protected] (C.L.E. Swartz). 0098-1354/$ – see front matter © 2013 Elsevier Ltd. All rights reserved. http://dx.doi.org/10.1016/j.compchemeng.2013.10.019

Poor SCM can contribute to unwanted instabilities within the network, such as the bullwhip effect, defined as the amplification in demand variability observed when moving up the supply chain from retailers to suppliers. The bullwhip effect was first illustrated through a series of case studies in the seminal work of Forrester (1961), and it has since been acknowledged that this major phenomenon is linked to forecast driven and decentralized decision-making, resulting in poor efficiency (Geary, Disney, & Towill, 2006; Lee, Padmanabhan, & Whang, 1997). Applications for SCM that integrate classical control technology have been motivated by the need for effective approaches to mitigate inefficiencies. These applications generally apply feedback laws for maintaining inventory positions and satisfying demand (e.g. Lalwani, Disney, & Towill, 2006; Perea-López, Grossmann, Ydstie, & Tahmassebi, 2001; Perea, Grossmann, Ydstie, & Tahmassebi, 2000; Towill, 1982). As discussed in Sarimveis, Patrinos, Tarantilis, and Kiranoudis (2008) in their comprehensive review on the application of control theory to SCM, the limitations associated with classical control technology, such as the inability to explicitly consider delays and interactions in the network, can be averted by applying an advanced control method such as model predictive control (MPC). Further advantages of MPC include a feed-forward control capability, an ability to address economics, and considerable flexibility in the underlying optimization formulation. MPC is a multivariable control method that has found wide application to industrial processes over the past three decades; however, the application of MPC to SCM has been considered more recently. Tzafestas, Kapsiotis, and Kyriannakis (1997) developed a generalized production planning framework utilizing the MPC

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Nomenclature Indicies/sets d∈D distribution site e, e ∈ E plant site inventory echelon j∈J material (chemical) raw material supplier ls ∈ LS m, m ∈ M plant site ps ∈ PS production scheme scenario s∈S t, t ∈ T time period (in supply chain model) t* ∈ T* actual time period JP set of final products PF set of products for FPM at plant site m Jm PI Jm set of products for IPM at plant site m JR set of raw materials RF Jm set of raw materials for FPM at plant site m RI Jm set of raw materials for IPM at plant site m I set of production schemes available at IPM at plant PSm site m F PSm set of production schemes available at FPM at plant site m Binary variables 1 if the IPM process unit in plant site m begins a prouIm,ps,t duction scheme ps at time period t; and 0 otherwise uFm,ps,t 1 if the FPM process unit in plant site m begins a production scheme ps at time period t; and 0 otherwise Continuous variables Bj,d,t quantity of back orders of final product j in distribution site d at time period t F quantity of final product j shipped from plant site m Fj,m,d,t to distribution site d at time period t IW Fj,m,t quantity of material transferred from intermediate product storage facility to warehouse in plant site m at time period t P Fj,e,e  ,m,m ,t quantity of material shipped from storage echelon e at plant site m to storage echelon e at plant site m in time period t S Fj,d,t quantity of final product j shipped from distribution site d to fulfil customer demand and back orders at time period t F Ij,m,t inventory of final product j at warehouse in plant site m at time period t I inventory of intermediate product j at intermediate Ij,m,t product storage facility in plant site m at time period t R Ij,m,t inventory of raw material j at raw material storage facility in plant site m at time period t S quantity of final product j inventory in distribution Ij,d,t site d at time period t F Pps,m,t quantity of main raw material which begins to undergo processing to final product in plant site m via scheme ps at time period t I quantity of main raw material which begins to Pps,m,t undergo processing to intermediate product in plant site m via scheme ps at time period t Oj,ls,m,t purchase quantity of raw material j to supplier ls from plant site m at time period t Parameters P ˇps process yield of product produced per unit of raw material consumed in production scheme ps

u

M m,ps l

M m,ps

ıM ps ıPm,m

ıRls,m ıSm,d

 Rls,m Fm,d Pm,m j,ps s ω1 ω2 R ˝m I ˝m F ˝j,m S ˝j,d F Dj,d,t

T n

maximum batch size for production scheme ps in plant site m minimum batch size for production scheme ps in plant site m manufacturing delay for production scheme ps (days) shipping delay between plant site m and m (days) delivery delay of procured material between supplier ls and plant site m (days) shipping delay between plant site m and distribution site d (days) ratio between weighting parameters (ω1 /ω2 ) maximum quantity of raw material which can be ordered from supplier ls during a time period maximum transportation capacity from plant site m to distribution site d during a time period maximum transportation capacity from plant site m to m during a time period mass balance coefficient of material j in production scheme ps probability of the occurrence of scenario s weighting parameter attributed to customer service (J1 ) weighting parameter attributed to operating cost (J2 ) maximum storage capacity of raw material in plant site m maximum storage capacity of intermediate product inventory in plant site m maximum storage capacity of final product j in plant site m maximum storage capacity of final product j in distribution site d (units) customer demand of final product j at distribution site d at time period t execution frequency of model predictive controller (days) length of prediction horizon (days)

approach, where decision variables include production, as well as advertising effort to influence sales. Bose and Pekny (2000) investigate the performance of a supply chain system where the demand level is uncertain. A hierarchical approach is proposed, where a forecasting model generates inventory targets (levels) to achieve a desired customer service level, and a scheduling model determines production tasks to achieve inventory targets, in a rolling horizon fashion as in MPC. Seferlis and Giannelos (2004) developed a two-layered control scheme for SCM, where a decentralized PID inventory controller is embedded within a MPC framework that computes shipments and places orders to nodes. Braun, Rivera, Flores, Carlyle, and Kempf (2003) implement a decentralized MPC framework for a supply chain in the semiconductor industry, and investigate control performance under plant model mismatch, and demand forecast error. Wang, Rivera, and Kempf (2007) examine the application of MPC as an integral component of a hierarchical framework for SCM within the semiconductor industry, where key challenges include high stochasticity in demand, and nonlinearity in model parameters. The effect of move suppression parameters, model parameters, and plant capacity on robustness and performance is illustrated through simulation case studies. Wang and Rivera (2008) enhance the prior study by formulating a multiple-degree-of-freedom observer for the systematic tuning

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of the MPC controller. They demonstrate that system robustness in the presence of uncertainty in demand, throughput times, and supply are achieved through proper tuning, and adequate capacity in the plant. Subramanian, Rawlings, Maravelias, Flores-Cerrillo, and Megan (2012) propose a novel cooperative MPC approach for coordination of supply chains, where each echelon makes its own local decisions, but the overall supply chain objective is optimized. The cooperative formulation guarantees closed-loop stability, and in simulation studies provides significantly superior performance over decentralized and noncooperative approaches. The supply chain model is a vital component of an MPC-based strategy, that must describe adequately the dynamic behavior of the system. A straightforward supply chain model captures inventory dynamics based on the quantity of material entering and leaving the echelon at each time period; however, a more sophisticated model addresses the hybrid nature of the supply chain. A hybrid system is a dynamic system that exhibits both continuous and discrete behavior. In a supply chain the discrete nature arises from the disjunctive logical conditions/rulesets governing decision making (e.g. production scheduling), generally formulated using integer variables, thus giving rise to a mixed-integer programming (MIP) formulation. As discussed in Mestan, Türkay, and Arkun (2006), SCM is a key application for the MPC of hybrid systems. Bemporad and Morari (1999a) propose a generalized mixed logical dynamical (MLD) framework for modeling and controlling systems described by linear dynamics and logical conditions. MPC is shown to effectively control a gas supply system, represented through the MLD form. Bemporad and Giorgetti (2006) describe a logic-based solution methodology applied for the optimal control of hybrid systems, and illustrate the applicability of the method for the control of a hybrid supply chain system with logical conditions that regulate production and distribution. Perea-López, Ydstie, and Grossmann (2003) developed a discrete-time mixed integer linear programming (MILP) model to describe the material and information (order) flow, and the continuous-discrete nature of the supply chain. MPC is applied for the centralized coordination of material, orders and production scheduling to maximize profit. In a similar study, Mestan et al. (2006) address the MPC of a hybrid supply chain system with multi-product manufacturing plants governed by logical conditions, that is modeled explicitly in the MLD form presented in Bemporad and Morari (1999a). Liu, Shah, and Papageorgiou (2012) present a MILP formulation for supply chain planning with sequence-dependent changeovers and price elasticity. They propose its implementation using an MPC approach that maximizes profit over the prediction horizon, with penalties on the total inventory deviation and product price change. Uncertainty is a relevant issue in SCO, and if not addressed can contribute to sub-optimal decision making. Multi-stage stochastic programming approaches have been proposed in the literature for the treatment of uncertainty in design, planning, and scheduling problems. In the multi-stage framework, some decisions are made “here-and-now” prior to the uncertain event taking place, and the remaining decisions are postponed in a “wait-and-see” manner after uncertainty has been resolved and more information is available. The objective is to optimize the deterministic first stage cost and the expectation of the second stage cost. As discussed in the recent review paper by Grossmann (2012), multi-stage stochastic programming is best suited when different recourse actions are possible depending on how the uncertainty resolves, which is essentially equivalent to a “quasi-simulation”. Two-stage stochastic programs are most common because multi-stage programs very quickly become intractable when the number of scenarios becomes large. You, Wassick, and Grossmann (2009) present a stochastic mid-term tactical planning model which captures uncertainty in demand and freight rates using a two-stage stochastic programming approach with incorporation of risk-management measures.

39

Through simulation they quantify an average cost savings to be expected when using the stochastic approach as compared to a deterministic approach. Balasubramanian and Grossmann (2004) consider the problem of scheduling a multi-product batch plant under demand uncertainty through a multi-stage stochastic programming formulation. However, to reduce the computational expense associated with solving a large scale multi-stage stochastic programming problem, an approximate solution strategy is proposed that relies on solving a number of smaller two-stage stochastic programming problems within a shrinking horizon framework. Optimization results indicate that the approximate strategy provides a solution within a few percent of the multi-stage result, at a significantly reduced computation time. The most common source of uncertainty addressed in SCO has been product demand. Demand forecasting through time series analysis (e.g. Seferlis & Giannelos, 2004), and holding excess inventory (safety stock) are methods proposed in the literature for hedging against uncertainty. An approach to address uncertainty explicitly at the execution of the model predictive controller is provided through robust MPC. The nominal MPC algorithm assumes in the input move calculation that the process to be controlled and the model used for prediction are the same and no unmeasured disturbance is acting on the process. In robust MPC, these assumptions are relaxed. When uncertainties are present, the future closed-loop behavior of the systems is uncertain, because the control action will depend on how the uncertainty has been resolved. An openloop approach to robust MPC computes a single control trajectory, neglecting the future corrective (recourse) actions of MPC itself. Without considering this partial compensation for uncertainty, the open-loop approach may be excessively conservative. An overview of robust model predictive control and a survey of relevant literature is given by Bemporad and Morari (1999b), where the concept of closed-loop prediction, among other concepts, is discussed in detail. Lee and Yu (1997) present robust MPC algorithms that minimize the worst-case cost (min–max) based respectively on open-loop and closed-loop prediction. The latter describes the future control action based on dynamic programming theory, where the control behavior for the worst-case performance is derived working backwards according the principle of optimality; however, the formulation is numerically demanding for even a moderate number of states and time periods. They compare the performance of robust MPC formulations based on closed- versus open-loop prediction. Kothare, Balakrishnan, and Morari (1996) present a robust MPC algorithm that minimizes an upper bound on a worst-case performance objective, and captures the closed-loop effects of the predicted response by computing a state feedback control law at each execution of the controller. The problem is formulated and solved using linear matrix inequalities (LMIs). Component-wise peak bounds on the inputs and outputs are included as sufficient LMI constraints. Kouvaritakis, Rossiter, and Schuurmans (2000) present a robust MPC scheme that relaxes the restriction of a constant state feedback control law by introducing an additional term in the control equation. This permits a reduction in on-line computation required, and improved performance during transients. Bemporad, Borrelli, and Morari (2003) propose a robust MPC strategy that considers unknown exogenous disturbances and parametric uncertainty. A min–max formulation is followed, and both open-loop and closed-loop prediction considered, the latter through a dynamic programming approach. A key feature of their approach is the use of multi-parametric programming to provide an explicit control law as a piecewise affine function of the state. Sakizlis, Kakalis, Dua, Perkins, and Pistikopoulos (2004) also present a multi-parametric programming formulation of robust MPC, where they consider additive time-varying uncertainty, and the expected performance of a quadratic objective over

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the uncertain domain. In a subsequent contribution, Pistikopoulos, Faísca, Kouramas, and Panos (2009) consider uncertainty in the state-space matrices in a multi-parametric robust MPC formulation. With parametric programming, the major computational expense is off-line; however the off-line computational effort can be significant. Warren and Marlin (2003) developed a robust MPC approach that approximates the future closed-loop behavior with unconstrained nominal MPC, and applies a probabilistic approach for capturing disturbance uncertainty. Through simulation studies, the proposed approach is shown to provide an improvement in dynamic performance over open-loop robust MPC and nominal MPC. Li and Marlin (2009) extend the prior work to consider a constrained nominal MPC controller for capturing the closed-loop behavior, which they formulate as a stochastic bi-level optimization problem that addresses model and disturbance uncertainty. The bi-level problem is approximated as a single level problem by assuming that a bound for a manipulated variable is either active or inactive for all realizations of the system. A “DMC” heuristic is applied to determine active bounds on manipulated variables. The proposed controller is applied for the control of a supply chain, and provides significant improvement in reducing the occurrence of back orders over a nominal MPC algorithm; however, the formulation has not been extended to address integer variables, typical for modeling decisions in the supply chain governed by logical conditions. This work extends MPC-based tools for SCM to explicitly address uncertainty in the control of hybrid supply chain systems. The approach presented in this paper addresses disturbance and model uncertainty explicitly, production scheduling in the plant governed by logical conditions/rulesets, and multiple supply chain metrics to evaluate performance, within an integrated optimization formulation. The optimization problem to be solved at each controller execution is formulated as a stochastic, bi-criterion, mixed-integer linear programming (MILP) problem. In this work, a scenario-based approach is applied for capturing uncertainty in the optimization formulation, and an additional nuance is the application of a stochastic forecasting model to generate demand scenarios. Furthermore, this paper demonstrates how two-stage stochastic programming can be applied for approximating the closed-loop prediction of uncertainty propagation, to compute less conservative control action than an open-loop prediction approach.

The remainder of the paper is organized as follows. Section 2 describes the supply chain system considered and the general problem statement. Section 3 describes the mathematical formulation of the dynamic model of the system. Section 4 presents the details of the open-loop and approximated closed-loop approach to robust MPC applied to SCM. Section 5 presents two case studies, where the robust MPC approaches developed are applied for controlling multi-product, multi-echelon supply chains. Finally, Section 6 concludes with some remarks and outlines future research avenues. 2. Problem statement We consider the supply chain system illustrated in Fig. 1, which was adapted from a case study originally presented in Li and Marlin (2009) and extended to address purchasing and manufacturing delays, production scheduling in the plant, and multiple raw materials, production schemes, and plant sites. An overview of the system follows. The box encompassing several echelons represents a plant site m. Each plant site includes the following echelons: raw material storage (RS), intermediate product manufacturing (IPM) and storage (IPS), final product manufacturing (FPM), and a warehouse (WH) for final product storage. A raw material purchase of Oj,ls,m,t units of material j is made to supplier ls from plant site m at time period t. The order arrives at the plant site after a delivery delay R units. of ıRls,m days. Raw material is stored with inventory of Ij,m,t

I A quantity of Pps,m,t units of main raw material is withdrawn from inventory and processed into intermediate product by the IPM echelon in plant site m via production scheme ps. The conversion of raw material into intermediate product is a single stage batch process, with a manufacturing delay of ıM ps days. Intermediate product I j is stored with inventory of Ij,m,t units. If material j acts as one of

P the raw materials at another plant site m , a quantity Fj,e,e  ,m,m ,t units is transferred from inventory echelon e of plant site m to echelon e of plant site m . If material j is one of the final products, a IW units is transferred from the IPS to the warehouse quantity Fj,m,t

F units of main raw material is in plant site m. A quantity of Pps,m,t withdrawn from the remaining inventory and processed further into final product j by the FPM echelon in plant site m via production scheme ps. The production of final product j from intermediate product is a single stage batch process, with a manufacturing delay

Fig. 1. Schematic of process supply chain system.

R. Mastragostino et al. / Computers and Chemical Engineering 62 (2014) 37–55

of ıM ps days. During the manufacturing duration of final product j, the production of an alternate product cannot begin at the FPM echelon. A similar restriction applies in the manufacture of intermediate F products. Final product j is stored with inventory of Ij,m,t units at the

F units is withdrawn warehouse in plant site m. A quantity of Fj,m,d,t from inventory and shipped from plant site m to distribution site d. The final product arrives at the distribution site after a delay of ıSm,d days, which reflects the transportation and material handling delay. Product within the distribution site which does not fulfill demand or accumulated back orders at time period t, is stored as “safety S stock” with inventory of Ij,d,t units. Safety stock is excess inventory held for hedging against uncertainty. The customer demand of final F product j at distribution site d at time period t is Dj,d,t . If sufficient inventory does not exist at the distribution site, product shortfall occurs, and the unfulfilled portion of demand is accumulated as a back order of Bj,d,t units. Back orders must be fulfilled before new demand requests can be satisfied. The objective of the SCO problem to be solved at each execution of the model predictive controller is to minimize system wide operating costs, while preventing back orders in the presence of uncertain demand and process yield. The following system assumptions are introduced:

(i) Raw materials are procured from the set of LS different suppliers. (ii) The IPM and FPM echelons in plant site m represent batch process units, which convert raw materials into intermediate products, and intermediate products into final products j within the set JP . (iii) Changeover times at manufacturing units are negligible in comparison to manufacturing times. (iv) Procurement, production, and transportation decision making occur at equivalent time intervals. It is worth mentioning that the formulation is flexible with regard to the supply chain system considered. Additionally, the supply chain model presented in the paper can be readily revised to relax assumptions, or add additional details (if needed). 3. Process supply chain model A discrete-time supply chain model is presented to describe the dynamic behavior of material and information flow within the supply chain. The discrete-time formulation divides the horizon into equal length intervals, T, where each time period is indexed by t. A discrete-time representation facilitates the inclusion of time delays (lags), and restricts decision making to occur at the beginning of each time period. The model is based on a material balance around each storage echelon in the supply chain.

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in plant site m at time period t is expressed in terms of the consumpI tion amount of its main raw material and is given by j,ps Pps,m,t , where j,ps is the mass balance coefficient of chemical j. Similarly, the production amount of intermediate product j in a production P PI task ps in plant site m at time period t is given by j,ps ˇps ps,m,t , P is the process yield of production scheme ps. The binary where ˇps variable uIm,ps,t is introduced to model a disjunction in the con-

I . Eqs. (4) and (5) restrict the consumption tinuous variable Pps,m,t l

I M ) to lie between a lower (m,ps amount of main raw material Pps,m,t Mu

and upper (m,ps ) bound, if uIm,ps,t is 1. uIm,ps,t is 1 if the IPM process unit in plant site m begins a production task ps at time period t; and 0 otherwise. The basic assignment constraints included in the model capture the logical conditions/rulesets that regulate production scheduling in the plant site. Similar constraints are proposed in Shah, Pantelides, and Sargent (1993), where the authors reformulated the assignment constraint originally derived in Kondili, Pantelides, and Sargent (1993) to improve computational performance. Eq. (6) is the full backward constraint that restricts the start of another production task ps at the IPM process unit at time period t, if a task has already begun within the backward interval [t − (ıM ps /T ) + 1, t].



R R Ij,m,t+1 = Ij,m,t +

Oj,ls,m,t−(ıR

ls,m

ls





m :m = / m

e

+



FP

m :m = / m

e

P Fj,rs,e  ,m,m ,t

/T )

RI ∀ j ∈ Jm , m, t

∀ j ∈ J R , ls, m, t

R R Ij,m,t ≤ ˝m

I j,ps Pps,m,t

I ps:ps ∈ PSm

m ,m



Oj,ls,m,t ≤ Rls,m



−

j,e ,rs,m ,m,t−(ıP 



−

/T )

∀ m, t

(1)

(2) (3)

RI j:j∈Jm u

I M Pps,m,t ≤ m,ps uIm,ps,t l

I M Pps,m,t ≥m,ps uIm,ps,t



t−(ıM ps /T )+1

I ps:ps ∈ PSm

t  =t



∀

m, ps ∈ PS Im , t

(4)

∀ m, ps ∈ PS Im , t

(5)

uIm,ps,t  ≤ 1 ∀ m, t

(6)

3.2. Intermediate product storage 3.1. Raw material storage The mass balance of raw material around the RS echelon in plant site m is given by Eq. (1). The delay of ıRls,m reflects the time between when an order for raw material j (Oj,ls,m,t ) is made to a supplier P and the corresponding delivery. The term Fj,e  ,e,m ,m,t represents the inter-plant shipment amount of chemical j from storage echelon e in plant m to storage echelon e in plant m with time delay of ıPm ,m . Product storage echelons rs, ips, and fps designate the raw material, intermediate product and final product storage echelons in inventory echelon set E. Eq. (2) represents the maximum order which can be made to supplier ls for material j (Rls,m ) during a time period. The total inventory of all raw materials at plant site m is restricted to a R , as given by Eq. (3). The quantity maximum storage capacity of ˝m of raw material j that begins to be consumed in a production task ps

The mass balance of material j around the IPS echelon in plant site m is given by Eq. (7). The total inventory of materials at IPS I as given by unit is restricted to a maximum storage capacity of ˝m IW Eq. (8). Fj,m,t represents the amount of material transferred from IPS to the warehouse in plant site m. The amount of chemical j generated or consumed in the production task ps at FPM process unit in plant site m at time period t is expressed in terms of the F of its main raw material. Eqs. (9) and consumption amount Pps,m,t F . (10) are required to model the disjunction in the variable Pps,m,t

The binary variable uFm,ps,t is 1 if the FPM process unit in plant site m begins a production task at time period t to produce final product through production scheme ps; and 0 otherwise. Eq. (11) is a full backward constraint for representing production scheduling at the FPM process unit, that restricts the start of another production task

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at the FPM unit at time period t, if another task has already begun within the backward interval [t − (ıM ps /T ) + 1, t].



I I Ij,m,t+1 = Ij,m,t +

P j,ps ˇps PI



+

m :m

= /

m

= /

m



−

m :m





−

FP

j,e ,ips,m ,m,t−(ıP 

m ,m

e



/T )

PF ∀ j ∈ Jm , d, t

PI RF ∀ j ∈ Jm ∪ Jm , m, t

F IW j,ps Pps,m,t − Fj,m,t

(7)

∀ m, t

(8)

PI ∪J RF j:j ∈ Jm m

S site m and distribution site d, and Fj,d,t is the quantity of final product j delivered from d at time period t to satisfy customer demand and accumulated back orders. Eq. (17) represents the back order balance for final product j at distribution site d.

S S Ij,d,t+1 = Ij,d,t +



FF

j,m,d,t−(ıS

m,d

m Mu

∀ m, ps ∈ PS Fm , t

F ≤ m,ps uFm,ps,t Pps,m,t F Ml ≥m,ps Pps,m,t

uFm,ps,t



t−(ıM ps /T )+1

F ps:ps ∈ PSm

t  =t



∀ m, ps ∈

PS Fm ,

uFm,ps,t  ≤ 1 ∀

(15)

The mass balance of final product j in distribution site d is given by Eq. (16), where ıSm,d is the transportation delay between plant

P Fj,ips,e  ,m,m ,t

e

I I Ij,m,t ≤ ˝m

(14)

j

3.4. Distribution site

F ps:ps ∈ PSm



∀ e, e , m, m , t

P P Fj,e,e  ,m,m ,t ≤ m,m

F F Ij,m,t ≤ ˝j,m

ps,m,t−(ıM ps /T )

I ps:ps ∈ PSm



/T )

S − Fj,d,t

∀ j ∈ J P , d, t (16)

(9)

t

(10)

∀ j ∈ J P , d, t

S F Bj,d,t+1 = Bj,d,t − Fj,d,t + Dj,d,t

(17)

The distribution sites operate with a “best I can do” policy (PereaLópez et al., 2003) indicated by Eq. (18),

m, t



(11) S Fj,d,t

=

S F Ij,d,t ≥Dj,d,t + Bj,d,t

F Dj,d,t + Bj,d,t ,

if

S Ij,d,t ,

S F if Ij,d,t < Dj,d,t + Bj,d,t

∀ j ∈ J P , d, t (18)

3.3. Warehouse The mass balance of final product j around the WH echelon within plant site m is given by Eq. (12). The shipment of products from plant site m to distribution site d is restricted by a maximum transportation capacity of Fm,d during a time period as given by Eq. (13). The inter-plant shipment quantity is restricted to a maximum quantity of P during a time period (Eq. (14)). Eq. (15) restricts the F . inventory of final product j to a maximum storage capacity of ˝j,m



F F Ij,m,t+1 = Ij,m,t +





m :m = / m

e

+

−





m :m = / m

e

FP

j,e,fps,m ,m,t−(ıP 

m ,m

−

F Fj,m,d,t ≤ Fm,d

/T )



S∗

upper bound on Ij,d,t , which reflects the maximum storage capacity of final product j at distribution site d, as given by Eq. (20).

IW + Fj,m,t ∗

∗

S S Ij,d,t+1 = Ij,d,t + F Fj,m,d,t



FF

j,m,d,t−(ıS

m,d

m

/T )

F − Dj,d,t

∀ j ∈ J P , d, t (19)

d

PF ∀ j ∈ Jm , m, t



∗

j,ps,m,t−(ıM ps /T )

P Fj,fps,e,m,m  ,t

∗

S S = Ij,d,t − Bj,d,t . Eqs. (16) the model through the substitution, Ij,d,t and (17) are then transformed into Eq. (19), where back orders S∗ is negative, and inventory of exist for final product j at d if Ij,d,t S is positive. We can now impose an final product j exists at d, if Ij,d,t

P ˇps j,ps P F

F ps:ps ∈ PSm

where all the demand and accumulated back orders at time period t are satisfied if sufficient stock is available; otherwise the available stock will be shipped. To capture this logical condition a binary variable is required; however, a construct was posed in Li and Marlin S (2009) to avoid additional integer variables by eliminating Fj,d,t in

(12)

∀ m, d, t

(13)

∗

S S ≤ ˝j,d Ij,d,t

∀ j ∈ J P , d, t

(20)

3.5. Delay representation

PF j:j∈Jm

The delays associated with procurement, manufacturing and shipping can be represented in an equivalent form by introducing additional state variables: K∗ = =

O FP PI PF FF   [(Kj,ls,m ∀ j, ls, m), (Kj,e,e  ,m,m ∀ j, e, e , m, m ), (Kj,ps,m ∀ j, ps, m), (Kj,ps,m ∀ j, ps, m), (Kj,m,d ∀ j, m, d)] O [[k1,j,ls,m , . . ., kO R (ı

ls,m

PI , . . ., kPIM [k1,j,ps,m

/ T ),j,ls,m

(ıps /T ),j,ps,m

FP FP ∀ j, ls, m], [k1,j,e,e  ,m,m , . . ., k P (ı

m,m

PF ∀ j, ps, m], [k1,j,ps,m , . . ., kPFM

/T ),j,e,e ,m,m

(ıps /T ),j,ps,m

T

∀ j, e, e , m, m ],

FF ∀ j, ps, m], [k1,j,m,d , . . ., kFFS (ı

m,d

/T ),j,m,d

∀ j, m, d]]

T

R. Mastragostino et al. / Computers and Chemical Engineering 62 (2014) 37–55

which denote the quantity currently on route or being processed. In the above, the indices corresponding to chemicals, suppliers, echelons, plant sites, production schemes and distribution centers would range over the sets over which they were defined in the SC model. These state variables are related to the original lagged variables at time t through an equation system shown below for raw material procurement,

⎛

⎞

O k1,j,ls,m,t+1

⎜ kO ⎜ 2,j,ls,m,t+1 ⎜ ⎜. ⎜ .. ⎝ kO R (ı

⎛

⎞

⎟ ⎜ kO ⎟ ⎟ ⎜ 3,j,ls,m,t ⎟ ⎟ ⎜ ⎟ ⎟ = ⎜. ⎟ ∀ j, ls, m, t ⎟ ⎜ .. ⎟ ⎠ ⎝ ⎠

/ T ),j,ls,m,t+1

ls,m

O k2,j,ls,m,t

where the lag terms in the model can now be replaced as given by Eq. (22). Oj,ls,m,t−(ıR

ls,m

/T )

=

O k1,j,ls,m,t

∀ j, ls, m, t

∗

S is positive. when Ij,d,t

∗

∀ j ∈ J P , d, t

S S ≥Ij,d,t + Bj,d,t Ij,d,t

∀ j ∈ J P , d, t

∗

S S R R I I F F [K ∗ , I1,1 , . . ., I|J|,|M| , I1,1 , . . ., I|J|,|M| , I1,1 , . . ., I|J|,|M| , I1,1 , . . ., I|J|,|D| ]

p=

S S [I1,1 , . . ., I|J|,|D| , B1,1 , . . ., B|J|,|D| ]

u=

I I F F [O1,1,1 , . . ., O|J|,|LS|,|M| , P1,1 , . . ., P|PS|,|M| , P1,1 , . . ., P|PS|,|M| ,

T

T

IW IW P P F F F1,1,1,1,1 , . . ., F|J|,|E|,|E|,|M|,|M| , F1,1,1 , . . ., F|J|,|M|,|D| , F1,1 , . . ., F|J|,|M| ,]

h=

[uI1,1 , . . ., uI|M|,|PS| , uF1,1 , . . ., uF|M|,|PS| ]

d=

F F [D1,1 , . . ., D|J|,|D| ]

T

T

T

4. Control problem formulation

S representing inventory in the distribution site (Ij,d,t ) to be non-zero

∗

∗

T

x=

(25)

Inventory variables and back order variables are non-negative as given by Eq. (23). Eqs. (24) and (25) force the back order variS∗ is negative and the variable able (Bj,d,t ) to be non-zero when Ij,d,t

S Bj,d,t ≥ − Ij,d,t

where x is a vector of the state variables, p is a vector of the auxiliary state variables, u is a vector of the continuous decision variables, h is a vector of the binary decision variables and d is a vector of the disturbance parameters, defined as,

(24)

(22)

3.6. Variable bounds

j, m, d, t

(28d)

The maximum dimension of the variables is indicated in the above. In practice, the variables would be defined over the sets indicated in the SC problem formulation, which in some cases are subsets of the full sets J, etc. Eq. (28a) represents the aggregation of Eqs. (1), (7), (12), (19) and (21), Eq. (28b) represents the aggregation of Eqs. (2)–(5), (8)–(10), (13)–(15), (20) and (23)–(26), and Eq. (28c) represents the aggregation of Eqs. (6) and (11). The parameter q represents the maximum manufacturing delay in the supply chain system minus 1, i.e. q = [max{ıM1 , . . ., ıM|PS| } − 1], c represents a vector of constants, and n is the prediction horizon (optimization horizon). Of particular importance, some of the binary variables in Eq. (28c) are undefined when the time period t is less than or equal to q, since they reflect time periods in the past. When solving the SCO problem at the execution of the model predictive controller, discrete decisions made in previous time periods (i.e. to begin a production task or not) influence decisions made at the current time period or in the future, so they are introduced as parameters in the optimization (i.e. ht is replaced by hfx ∀ t = −(q + 1), . . ., 0), t where hfx t is a vector of parameters. A similar representation of a supply chain model is proposed in Mestan et al. (2006), adapted from the MLD form presented in Bemporad and Morari (1999a).

Similar equations are introduced to represent the delays associated with manufacturing and shipping. A similar construct has been used in Li and Marlin (2009), which allows representation of the model in state-space form, a convenient structure for MPC and analysis.

R I F S Ij,m,t , Ij,m,t , Ij,m,t , Ij,d,t , Bj,d,t ≥0 ∀

ht ∈ {0, 1}(2×|PS|×|M|) t = 1, . . ., n

(21)

Oj,ls,m,t

43

(23)

S Eqs. (24) and (25) are valid if Ij,d,t and Bj,d,t are minimized in the objective function of the SCO problem, and have the effect of setting S ∗ and I S S ∗ is negative, and B S Bj,d,t = −Ij,d,t = 0 if Ij,d,t j,d,t = 0 and Ij,d,t = j,d,t

4.1. Nominal MPC

and binary decision variables can take a value of 1 or 0 as given by Eq. (27),

The primary objective of MPC is to determine the trajectory of future inputs that optimizes a performance criterion over a specific prediction horizon. Once the optimal input trajectory is computed, only the first control action is implemented on the process at the current time period. This reflects a key benefit over open-loop optimization, because at the next sampling instance, new state information is available from the process, and the optimal input trajectory is re-computed. In the presence of plant-model mismatch and unmeasured disturbances, the current solution trajectory is likely no longer an optimal or even feasible solution for subsequent time periods.

uIm,ps,t , uFm,ps,t ∈ {0, 1} ∀ m, ps ∈ PS Im ∪ PS Fm , t

4.2. Performance function

∗

∗

S S if Ij,d,t is positive. The continuous decision variables associated Ij,d,t with ordering, production and shipment are non-negative as given by Eq. (26),

I F P F IW Oj,ls,m,t , Pps,m,t , Pps,m,t , Fj,e,e  ,m,m t , Fj,m,d,t , Fj,m,t ≥0

∀ j, ls, ps, m, m , e, e , d, t

(26)

(27)

3.7. State-space representation The supply chain model presented can be equivalently represented as a state-space model of the form, xt+1 = A1 xt + B1 ut + Gdt t = 1, . . ., n − 1

(28a)

A∗1 xt + A∗2 pt + B1∗ ut + B2∗ ht + G∗ dt + c ≤ 0 t = 1, . . ., n

(28b)

E1 ht + E2 ht−1 + . . . + Eq ht−q ≤ 1 t = 1, . . ., n

(28c)

A number of quantitative metrics exist for evaluating the performance of a supply chain, a comprehensive review of which is given in Beamon (1999). Two key criteria considered here are an evaluation of economics and customer service. Eq. (29) represents the total summation of back orders, denoting a measure of customer service. Eq. (30) represents the total operating cost in terms of the variables defined in the state-space model, where Cx is a vector of cost coefficients for held inventory in the plant, Cp is a vector of cost coefficients for held inventory in the distribution site, and Cu is a vector of costs associated with the decision variables (i.e. raw

44

R. Mastragostino et al. / Computers and Chemical Engineering 62 (2014) 37–55

where ω1 and ω2 are the weighting parameters for each performance metric. The ratio of ω1 to ω2 is a tunning parameter in the optimization, defined here as  : = ω1 /ω2 . Finally, it is worthwhile to note that the performance function (J∗ ) is linear, thus giving rise to a linear MPC framework.

The uncertainty in customer demand is characterized by the uncertain elements within the disturbance vector dt . Demand uncertainty is resolved after decisions in the current time period are computed, i.e. after the execution of the model predictive controller. Uncertainty in the process yield of intermediate and final P ) is characterized product within the manufacturing sites (i.e ˇps by the uncertain elements within the matrix B1 in the state-space model. The yield can be thought of as an end-point quality of the batch (resolved after the batch operation is complete). The uncertainty in yield is resolved after the batch is complete. We consider the current time period for which a control action needs to be computed to correspond to t = 1, with the prediction horizon extending to t = n. As discussed earlier, the production constraints use a backward time formulation that involves binary variables corresponding to a number of previous time periods. These discrete inputs, ht for t < 1, are known and treated as parameters, denoted as hfx in Eqs. (32d)–(32f). At the end of the control t−k period, T, the control horizon shifts relative to the actual time period, denoted by t* , with the controller time periods again running from 1 to n.

4.3. Open-loop approach to robust MPC

4.4. Approximated closed-loop approach to robust MPC

We fist develop an open-loop approach to robust MPC, which will be compared with the closed-loop approach presented in the subsequent section. The open-loop approach, denoted here as ROF (robust open-loop formulation), utilizes a stochastic supply chain model and disturbance parameter to predict future system behavior. A scenario-based approach is applied to capture the uncertainty in product demand and process yield. A control trajectory is computed that is robust for all scenarios and the entire control trajectory is computed before uncertainty is resolved (i.e. no recourse action). The ROF is given by,

The ROF proposed in Section 4.3 may result in overly conservative control action causing excess safety stock, because in actual operation the effect of uncertainty is partly mitigated by feedback. To rigorously model the future closed-loop behavior, a multi-stage stochastic approach can be applied. The multi-stage approach mimics the following decision making procedure: a decision is made before an uncertain event occurs, then after the uncertainty is resolved and more information is available a recourse (corrective) decision is made; however decisions made in previous stages can not be changed. When decision making occurs over multiple time periods, as in SCM, uncertainty is revealed sequentially over time, which lends itself naturally to a multi-stage stochastic formulation. With a multi-stage approach, the future control action can be captured, because decisions computed in future time periods are in response to how the states have been propagated by the resolved uncertainty. However, the multi-stage formulation becomes computationally expensive and intractable when the number of time periods and scenarios becomes large. To alleviate the computational expense, a two-stage stochastic framework is presented to approximate the future closed-loop behavior. This implies that decisions made in the first time period occur before uncertainty is realized (i.e. first stage), and the decisions within the remaining time periods are postponed until after uncertainty is resolved (i.e. second stage). This is described as an approximate closed-loop approach because decisions computed after the first time period no longer hedge against the possibility that another scenario can resolve; however, with MPC only the decisions for the first time period (first stage) are implemented. The closed-loop approach to robust MPC, denoted here as RCF (robust closed-loop formulation) is given by,

material procurement, production and transportation). Operating cost denotes a measure of economic performance. J1 :=

n−1 

⎡ ⎣

⎤

Bj,d,t+1 ⎦

(29)

j:j ∈ J P ,d

t=1

J2 :=



n−1  

CxT xt+1 + CpT pt+1 + CuT ut



(30)

t=1

The multi-objective optimization problem is solved by applying the weighted-sum method indicated by Eq. (31), J∗ := ω1 J1 + ω2 J2

min ut , ht

s.t.

E[J∗ ] :=

(31)



s J∗s

(32a)

s

xt+1,s = A1 xt,s + B1s ut + Gdt,s

∀ s, t = 1, . . ., n − 1 (32b)

A∗1 xt,s + A∗2 pt,s + B1∗ ut + B2∗ ht + G∗ dt,s + c ≤ 0 ∀

s, t = 1, . . ., n (32c)

fx E1 ht + E2 hfx t−1 + . . . + Eq ht−q ≤ 1 t = 1

(32d)

fx E1 ht + E2 ht−1 + E3 hfx t−2 + . . . + Eq ht−q ≤ 1 t = 2

(32e)

.. . E1 ht + E2 ht−1 + . . . + Eq−1 ht−q+1 + Eq hfx t−q ≤ 1

E1 ht + E2 ht−1 + . . . + Eq ht−q ≤ 1 t = q + 1, . . ., n ht ∈ {0, 1} x1,s = x˜

(2×|PS|×|M|)

∀ s

t=q

(32f)

(32g)

min

t = 1, . . ., n

∗

E[J∗ ] := J(1) +



(2)

where E[J∗ ] represents the expectation of the dynamic performance, s refers to the scenario resolved, s is the probability of scenario s occurring, n is the length of the prediction horizon, and x˜ represents the initial value of the state variables. Optimizing the expectation of the dynamic performance is typically less conservative than optimizing the worst-case performance (i.e. min–max) used in some robust MPC formulations.

(33a)

s

(32h) (32i)

(2)∗

s Js

(1)

(1)

s.t. xt+1,s = A1 xt,s + B1s ut (2)

(2)

(2)

xt+1,s = A1 xt,s + B1s ut,s + Gdt,s A∗1 xt,s + A∗2 pt,s + B1∗ ut (1)

(1)

(1)

∀ s, t = 1

(33b)

∀ s, t = 2, . . ., n − 1

(33c)

+ Gdt,s

+ B2∗ ht

(1)

+ G∗ dt,s + c ≤ 0 ∀ s, t = 1 (33d)

R. Mastragostino et al. / Computers and Chemical Engineering 62 (2014) 37–55

appropriate model; however this aspect is beyond the scope of this work.

A∗1 xt,s + A∗2 pt,s + B1∗ ut,s + B2∗ ht,s + G∗ dt,s + c ≤ 0 (2)

(2)

(2)

(2)

∀ s, t = 2, . . ., n (1)

E1 ht

(33e)

fx + E2 hfx t−1 + . . . + Eq ht−q ≤ 1 t = 1

(2)

(33f)

(1)

fx E1 ht,s + E2 ht−1 + E3 hfx t−2 + . . . + Eq ht−q ≤ 1 ∀ s, t = 2

(2)

(2)

(1)

(2)

(2)

(2)

(1)

E1 ht,s + E2 ht−1,s + . . . + Eq−1 ht−q+1,s + Eq ht−q ≤ 1 ∀ s, t = q + 1 (33i) (2)

(2)

(2)

E1 ht,s + E2 ht−1,s + . . . + Eq ht−q,s ≤ 1 ∀ s, t = q + 2, . . ., n (33j) ∈ {0, 1}(2×|PS|×|M|)

(2)

ht,s ∈ {0, 1}(2×|PS|×|M|) (1)

t=1

(33k)

∀ s, t = 2, . . ., n

(33l)

∀ s

x1,s = x˜

J

:=

(1) ω2 [CuT u1 ]

(34)

(2)∗ Js

and represents the second stage performance objective for scenario s, defined as,

∗

(2)

Js

:= ω1 J1,s + ω2

The scenario-based approach has been regularly applied in optimization for capturing uncertainty in terms of a finite number of discrete realizations of stochastic parameters. Furthermore, scenario-based approaches allow for flexible uncertainty representations, since the underlying structure of the model is unchanged and independent of how scenarios are generated. A Monte Carlo sampling method is applied for generating scenarios (Hammersley & Handscomb, 1964). The Monte Carlo method entails generating a discrete set of scenarios by sampling from the continuous probability distribution, where the complete realization of all uncertain parameters in the model gives rise to a scenario. The forecasting model given by Eqs. (37) and (38) is used to generate demand scenarios,

∀ j ∈ J P , d, s, l = 0

F Dj,d,t ∗ +l,s =

F Dj,d,t ∗ +l−1 + aj,d,t ∗ +l,s

F Dj,d,t ∗ +l,s =

F Dj,d,t ∗ +l−1,s + aj,d,t ∗ +l,s − j,d aj,d,t ∗ +l−1,s

(37)

(33m)

where superscript (1) denotes a first stage variable, superscript (2) denotes a second stage variable, and represents the decision ∗ (1) (1) (2) (2) variables: ut , ht , ut,s , and ht,s for t = 1, . . ., n. J(1) represents the first stage performance objective defined as, (1)∗

(36)

4.6. Scenario generation

E1 ht,s + . . . + Eq−2 ht−q+2,s + Eq−1 ht−q+1 + Eq hfx t−q ≤ 1 ∀ s, t = q (33h)

(1)

F F F ∇ Dj,d,t := Dj,d,t − Dj,d,t−1 = aj,d,t − j,d aj,d,t−1 ∀ j ∈ J P , d, t

(33g)

.. .

ht

45

 n−1 

(2)

(2)

n−1   

CxT xt+1,s + CpT pt+1,s +

t=1

(2)

CuT ut,s





t=2

(35) with J1 as defined in Eq. 29. It is important to mention that Eq. (28c) in the supply chain model includes a summation of first and second stage variables from time period t = 2 to t = q + 1, as reflected in Eqs. (33g)–(33i) in the formulation. 4.5. Demand forecast model Demand forecasting using time series models is presented in the literature for improving control performance in the presence of demand uncertainty (e.g. Seferlis & Giannelos, 2004; Wang et al., 2007). In this work, the nonstationary integrated moving average model indicated by equation Eq. (36) is applied to forecast the stochastic disturbance (demand) over the prediction horizon n. ∇ is a backward difference operator,  j,d is a model parameter and aj,d,t is a white noise process described by a normal distribution with zero 2 . Models of this kind have often been found mean and variance of j,d useful in representing some commonly occurring time series, and to represent certain kinds of disturbances occurring in industrial processes and in econometrics (Box & Jenkins, 1970). It is worth mentioning that the framework is flexible enough to consider other time series models for demand forecasting. Additionally, the accuracy of the prediction will depend on the identification of an

∀ j ∈ J P , d, s, l = 1, . . ., n − 1

(38)

where l is the forecast lead time, and the input aj,d,t ∗ +l,s is sampled from a normal distribution, a∼N(, 2 ). The demand forecast depends on the demand resolved at the previous time period (t* − 1) as illustrated by Eq. (37). This strategy addresses the autocorrelation and moving average nature typically apparent with demand. This is expected to lead to less conservative control action than assuming independent demand uncertainty at each future time period. We consider in this article process yield of intermediate and final product j at each plant site m to be approximated by a normal distribution. Independent process yield scenarios are generated by sampling from the continuous distribution. Each scenario s represents an outcome of both the demand trajectory and process yield parameters of the model. Each scenario is then assigned an equivalent probability of occurrence, with the summation of probabilities for all scenarios equal to 1, i.e. s = 1/ns, where ns is the number of scenarios generated for capturing uncertainty. The objective function, E[J∗ ], of the supply chain optimization formulation represents a Monte Carlo estimator of the true expected value of J∗ (Liu & Sahinidis, 1996).

4.7. Closed-loop implementation Fig. 2 illustrates how the MPC strategies are implemented in closed-loop. The steps are as follows: (i) generate yield and demand scenarios to apply within SCO problem, (ii) solve SCO problem at each execution of the model predictive controller to compute control action (iii) implement control action for current time period on process/simulation model, and (iv) at next sampling period feed back updated state information to model predictive controller. We have assumed in this work that the execution frequency of the model predictive controller is T (i.e. same as model discretization); however this assumption can be readily relaxed.

46

R. Mastragostino et al. / Computers and Chemical Engineering 62 (2014) 37–55 Table 2 Size of supply chain optimization formulation for Case Study I (150 scenarios applied to capture uncertainty in robust MPC frameworks).

Fig. 2. Implementation of robust MPC strategy in closed-loop.

5. Case studies To demonstrate the robust MPC approaches in the previous section, we consider two case studies. The first involves a simple configuration, and compares the performance of nominal MPC and two robust MPC formulations. A pareto analysis between economics and customer service is also presented, and the effect of disturbance and model uncertainty is explored. The second case study illustrates the application of the MPC formulations to a more complex configuration involving two suppliers, two final products, and three intermediate products. 5.1. Case Study I 5.1.1. Case study description The ROF and RCF proposed are applied for the control of a multiechelon supply chain, with a single supplier (LS1 ), plant site (M1 ), and distribution site (DC1 ). The plant site is dedicated to producing an intermediate product and a single final product that is sold to customers. Table 1 summarizes the parameter values used in the case study presented. The production scheme PS1 processes raw material A to intermediate product B at the IPM unit, and production scheme PS2 converts B to final product C at the FPM unit. We compare in this paper the performance with the ROF and RCF to the performance with a nominal MPC framework, denoted by NOF (nominal open-loop formulation), that does not explicitly Table 1 Parameter values used in the Case Study I.a Parameter

Value

Parameter

Value

n (days)  Variance ( 2 ) of a Range of DF R (units) ˝M

15 0.1 9 0–50 500

Simulation length (days) Range of yield parameters No. of scenarios T (days) I ˝M (units)

40 0.45–0.95 150 1 500

1

S ˝C,D (units)

RLS

1

1 ,M1

(units)

P (units) ıRLS ,M (days) 1

1

ıM (days) PS

300

1

F ˝C,M (units)

500

(units)

100

(days)

4

120

FM

– 3

ıSM

1

1 ,D1

1 ,D1

1

2

ıP (days) Mean () of ˇPS1

ıM (days) PS

– 0.8

Cost of OA , in Cu ($/unit) I Cost of PPS , in Cu ($/unit)

0.5 1

Mean () of ˇPS2

0.75

Variance ( 2 ) of ˇPS1 Variance ( 2 ) of ˇPS2 Mu M (units) ,PS1

0.0025 0.0025 120

Cost of FCF , in Cu ($/unit) Cost of FP , in Cu ($/unit) Cost of IAR , in Cx ($/unit)

0.15 – 1

M M

(units)

25

Cost of IBI , in Cx ($/unit)

3.2

M M

(units)

150

Cost of ICF , in Cx ($/unit)

1.2

(units)

30

Cost of ICS , in Cp ($/unit)

1.5

1

1 l

1 ,PS 1 u

1 ,PS 2 Ml M 1 ,PS 2

a

2

1

F Cost of PPS , in Cu ($/unit) 2

Values of cost vector elements not specified in the table are zero.

Framework

Continuous variables

Discrete variables

NOF ROF RCF

324 35,935 47,557

28 28 3,902

account for uncertainty. In the NOF, uncertain model parameters are assigned average values, and the demand forecast within the prediction horizon is equated to the demand resolved at the previous time period. In the ROF and RCF, uncertain model and disturbance parameters are described by multiple scenarios with equal probability of occurrence, generated using the method presented in Section 4.6. 150 scenarios are used in this case study in the controller optimization formulation. The “actual” supply chain system is represented by a simulation model that reflects a particular outcome of the uncertain model parameters. Furthermore, a demand trajectory is computed from the forecast model which represents the “actual” demand resolved. Each closed-loop case is simulated, as depicted in Fig. 2, for a 40-day period. The performance of the closed-loop simulation is compared in terms of the supply chain metrics in the objective function of the supply chain optimization (i.e. operating cost and summation of back orders), as well as an alterative metric of customer service, which indicates the percentage of demand filled immediately, denoted as fill rate (FR) (Beamon, 1999). The SCO problem is modeled with GAMS 23.7.3 and solved using CPLEX 12.3 with parallel mode enabled (8 threads) to a 1% optimality gap. Simulations were performed on a 2.93 GHz Intel® CoreTM i7 DELL Vostro machine with 8 GB of RAM, running Windows 7 Professional 64-bit. Table 2 summarizes the model size of each optimization framework implemented. As observed from the table, the number of discrete and continuous variables is larger in the RCF, as compared to the ROF and NOF, since second stage decision variables are dependent on the number of scenarios considered. 5.1.2. Pareto analysis For supply chain systems, a tradeoff exists between economics and customer service; therefore the ratio between the weighting parameters, i.e.  : = ω1 /ω2 , can be thought of as a tuning parameter in the SCO problem. A large  represents more weight on minimizing back orders, while small  represents more weight on minimizing operating cost. Fig. 3 illustrates the Pareto trade off curve (frontier) between supply chain metrics. The curve was

1.125

Fig. 3. Pareto relationship between operating cost (J2 ) and customer service (J1 ) with ROF for Case Study I (× — J1 ; + - - FR). Note: Each data mark corresponds to average performance for 50 independent closed-loop simulations.

R. Mastragostino et al. / Computers and Chemical Engineering 62 (2014) 37–55

generated with the ROF, by performing a number of closed-loop simulations at different values of , and each data mark reflects the average performance over 50 independent simulations performed. Each closed-loop simulation, among the 50, represents a different resolution of demand and process yield uncertainty. The 50 closedloop simulations that generate each mark are repeated 3 times; the set of 50 parameter realizations used to represent the actual plant and demand trajectory is the same for each of the 3 batteries of closed-loop simulations. The repetitions do not produce identical results because at each execution of the model predictive controller the scenarios generated for capturing uncertainty in the optimization are not equivalent, leading to different control actions computed. However, the relatively close agreement between repetitions indicates that a statistically sufficient number of scenarios has been applied for capturing uncertainty in the optimization formulation. Scenario reduction using the method presented in You et al. (2009) can potentially improve computational performance; however this aspect has not been considered in the current work. On the region of the Pareto curve between locations (1) and (2), a large increase in operating cost, J2 , is accompanied by a minimal increase in the customer service (minimal decrease in back orders, J1 ). From an economic perspective, operating the supply chain at the point on the Pareto curve denoted by (1) is superior to (2), because comparable customer service is achieved at a significantly lower operating cost. For the region to the left of (1), an exponential increase in back orders is evident, with a decrease in operating cost, since less safety stock is held to hedge against uncertainty. It is clear that  has an effect on the conservativeness of the control action.

The trend in fill rate (FR) agrees with the trend in J1 , i.e an increase in fill rate corresponds to an improvement in customer service. The Pareto analysis is significant for determining the optimal operating region, i.e. the optimal value of the tunning parameter  to apply in the optimization. Of significance is that within different supply chain environments, customer tolerances for back orders are different, implying that the decision-maker can select a value of  dependent on operational preferences. 5.1.3. Comparing closed-loop performance with ROF, RCF and NOF 5.1.3.1. Graphical comparison for single outcome. The performance of the closed-loop simulation, with the ROF, RCF and NOF are compared graphically for the same particular outcome of uncertainty, i.e. same simulation model and demand trajectory (DF ). Figs. 4–6 illustrate the simulated closed-loop behavior of the supply chain system with the ROF, RCF and NOF. Since this case study involves a single raw material, intermediate product, and final product, subscript designations of chemicals associated with inventories, orders, and demand are not needed, and omitted. In this particular closed-loop simulation a value of 100 is assigned to . Fig. 4 illustrates the closed-loop result with the NOF. When uncertainty is not addressed in the optimization, significant product back orders accumulate, particularly after day 11, as illustrated by the trajectory of back orders, B, in Fig. 4. Inventory of final product in the warehouse of the plant site (IF ) and the distribution site (IS ) is driven to near zero, resulting in insufficient safety stock to remain robust against uncertainty; particularly we can observe that significant back orders accumulate after the increase in demand occurring after day 10.

II

O

100

50 10

20

30

10

20

30

10

20

30

10

20

30

10

20

30

10

20

30

100

40 20 0

50 0

40

PI

IR

100

0

47

10

20

30

50 0

40

150

PF

IF

100 50 0

100 50

10

20

30

0

40

100

FF

IS

40 20 0

10

20

30

DF

B 50 0

0

40

100

10

20

30

40

50

40 20 0

Time (day) Fig. 4. Closed-loop result with NOF – Case Study I.

Time (day)

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R. Mastragostino et al. / Computers and Chemical Engineering 62 (2014) 37–55

O

50 0

II

100

80 60 40 20 0

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IR

100

10

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30

50 0

40

150

0

IS

PF

50

60 40 20 0

10

20

30

0

40

60

10

20

30

0

40

DF

B

40 20

100 50 0

100 50

FF

IF

100

10

20

30

40

40 20 0

Time (day)

Time (day)

Fig. 5. Closed-loop result with ROF – Case Study I.

Fig. 5 illustrates the closed-loop result with the ROF. When addressing uncertainty at each execution of the model predictive controller, safety stock is maintained throughout the supply chain system to adequately respond to an increase in demand. We can observe from Fig. 5 that a minimal amount of back orders accumulates at day 22; however, they are quickly fulfilled within a day or two. There is a substantial improvement in customer service, illustrated from the trajectory of back orders (B), with the ROF, as compared to the NOF. Fig. 6 illustrates the closed-loop result with the RCF. We anticipate that the RCF will give less conservative control action, because future closed-loop behavior is approximately accounted for. The total operating cost over the simulation period with the RCF is 1.168 × 104 , as compared to 1.467 × 104 with the ROF, which reflects a 20.4% decrease in operating cost. It is important to acknowledge that with the RCF slightly more back orders accumulate from less conservative control action; however, the resultant customer service performance is still superior as compared with the NOF. 5.1.3.2. Comparison between multiple simulations. The previous section compares the closed-loop system behavior for a single outcome of the uncertain parameters. To gain more perspective on the system behavior when implementing the ROF, RCF and NOF, a number of closed-loop simulations are performed. Three cases are investigated, where each case differs in the value of  implemented. The simulation results are summarized in Table 3, and represent the average behavior of the system for 50 independent closed-loop simulations. As indicated by the results, a significant difference in performance is apparent between the robust and

nominal frameworks. The average summation of back orders with the NOF is significantly larger, indicating a substantial improvement in customer service with the ROF and RCF. The average fill rate with the NOF is less than 0.5 for each case, as compared to over 0.9 with the ROF and RCF, indicating consistency between customer service metrics. As expected, the results indicate that  has no appreciable effect on the closed-loop performance with the NOF. It must be emphasized that while the improvement in customer service is prominent when applying the robust MPC frameworks for SCM, the operating cost is larger because safety stock is maintained to hedge against uncertainty. The key objective is to not maintain an overly conservative quantity of safety stock. The results reported in Table 3 indicate a reduction in operating cost with the RCF versus the ROF, accompanied by somewhat poorer customer service. However, the level of customer service is still significantly superior compared with the NOF. Table 3 indicates the average computation time in seconds to solve the SCO problem at each execution of the controller. As observed from the table, although the average computation time is larger with the RCF, it is still reasonable, particularly for SCM applications where the execution frequency of the controller is in the order of days. 5.1.3.3. Effect of fine tuning parameter . As seen in Table 3, the same values of  result in different values of the performance objectives for the ROF and RCF strategies. Here, we compare the economic performance of the ROF and RCF approaches at an equivalent level of customer service, by generating and comparing the Pareto curves for the two strategies. We also investigate the sensitivity of the performance to the ratio .

80 60 40 20 0

10

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30

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40

10

20

30

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30

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40

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II

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50

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49

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R. Mastragostino et al. / Computers and Chemical Engineering 62 (2014) 37–55

10

20

30

0

40

150 50 0

100

PF

IF

100

50 10

20

30

0

40

40

80 60 40 20 0

FF

IS

60 20 0

10

20

30

40

50 0

40

DF

B

100

10

20

30

20 0

40

Time (day)

Time (day)

Fig. 6. Closed-loop result with RCF – Case Study I.

Fig. 7 illustrates the comparison between Pareto curves. It is important to note that different values of  are required to generate comparable levels of customer service. As indicated by locations (1) and (2) on the figure, a value of 75 is used for  with the RCF, compared to a value of 25 with the ROF to achieve a comparable level of customer service. The Pareto curve generated with the RCF is located to the left of the curve generated with the ROF, i.e. the economics are more favorable at an equivalent level of customer service by ∼2−4 %. With the scale of supply chain networks, a few percent in improvements could amount to a large dollar value in savings. At high levels of customer service, the Pareto curves converge; however, at even slightly reduced levels of customer service, simulation results indicate that the RCF performs more favorably. We have shown that  has a significant effect on the performance; we now consider the sensitivity of the performance to the ratio . Fig. 8 illustrates the sensitivity of  on the level of customer service, i.e. accumulation of back orders (J1 ), with the RCF and ROF. The shaded region on the figure represents the range we designate as acceptable customer service performance. The region outside the acceptable range is overly conservative (below) or not

sufficiently robust (above). The “acceptable region” is subject to the supply chain environment and operating preference. As observed from the figure, the ROF is more sensitive to the ratio, evident by the exponential trend versus linear trend with the RCF. This implies that a decrease in  (less emphasis on maximizing customer service) results in a more rapid degradation in performance with the ROF as compared with the RCF. Consequently, the range of  for the customer service to remain in the acceptable region is significantly larger with the RCF, as evident in the figure (i.e. ϕ1 > ϕ2 ). 5.1.3.4. Effect of disturbance and model uncertainty. In this section the influence of uncertainty type is illustrated. The closed-loop simulation results are summarized in Table 4, where each case differs in whether demand or yield is certain or uncertain. As evident by the results, demand uncertainty has the dominant effect on the occurrence of back orders, indicated by the much lower quantity accumulated when it is a certain parameter; however uncertainty in yield still has a noticeable effect. We expect the accumulation of back orders (J1 ) to increase when demand and process yield is uncertain (case B), as compared to when only demand is uncertain

Table 3 Summary of results with RCF, ROF and NOF for multiple closed-loop simulations – Case Study I. Framework Case,  J1 J2 × 10−4 FR CPU (s)

NOF 100 819 0.784 0.472 0.136

75 819 0.784 0.472 0.137

ROF 50 819 0.784 0.472 0.139

100 6.42 1.363 0.986 1.33

Note: Results correspond to the average performance for 50 independent closed-loop simulations.

75 12.6 1.327 0.982 1.32

RCF 50 11.1 1.266 0.979 1.32

100 54.2 1.127 0.950 10.2

75 69.8 1.089 0.935 10.2

50 98.5 1.046 0.920 11.2

50

R. Mastragostino et al. / Computers and Chemical Engineering 62 (2014) 37–55 Table 5 Size of supply chain optimization formulation for Cast Study II (50 scenarios applied to capture uncertainty in robust MPC frameworks). Framework

Continuous variables

Discrete variables

NOF ROF RCF

956 16,789 38,321

58 58 2,753

uncertainty in process yield in the optimization problem magnifies the conservativeness of the ROF, while this is not observed with the RCF. 5.2. Case Study II

Fig. 7. Comparing Pareto curve generated with ROF and RCF for Case Study I (× — ROF; +- - RCF). Note: Each data mark corresponds to average performance for 50 independent closed-loop simulations.

5.2.1. Case study description The second case study illustrates the application of the ROF and RCF control approaches to more complex systems and considers a multi-product, multi-echelon supply chain comprising of two suppliers (LS1 , LS2 ), two plant sites (M1 , M2 ) with a total of four production schemes (PS1 , . . ., PS4 ), and two distribution sites (DC1 , DC2 ). There are 4 different production schemes available to produce two final products E and G from two raw materials A and B. Materials C, D, and F are intermediate products. Production schemes PS1 and PS3 are available at the intermediate production facility while production schemes PS2 and PS4 are available at the final production facility. An intermediate product (here E) can be treated as a final product, and therefore the formulation allows the intermediate product to ship to the distribution center through the warehouse. Plant site M2 requires the intermediate product D as one of the raw materials for the production scheme PS3 , and thus it can not start the production until it receives the material D from plant site M1 . Case study data are summarized in Appendix A (Tables A.1–A.11). The production schemes considered are as follows: PS1 PS2 PS3 PS4

Fig. 8. Comparing the sensitivity of the ratio between weighting parameter () to the closed-loop customer service performance with ROF and RCF for Case Study I (× — ROF; +- - RCF). Note: Each data mark corresponds to average performance for 50 independent closed-loop simulations.

(case D). This is observed with the NOF and RCF; however with the ROF, J1 decreases when uncertainty in demand and process yield are considered versus just demand. We suspect that the conservative nature of the ROF is the cause of this inconsistency, since the operating cost (J2 ) increases substantially, i.e. ∼24%, from case D to case B as compared to only ∼2% from case D to B with the RCF. When considering both yield and demand uncertainty with the ROF, the increased safety stock held at the plant site as a result of considering yield uncertainty likely has a positive effect on mitigating demand uncertainty. Therefore, it would seem that considering

(M1 ) : (M1 ) : (M2 ) : (M2 ) :

A + B −→ C + D C −→ 0.5 E + 0.5 G A + D −→ E + F F −→ G

We restrict the inter-plant shipments to originate from the intermediate product or final product storage locations, and to terminate at the raw material or intermediate product storage locations. As described in Case Study I, NOF simulation runs are assigned average values of uncertain model parameters and demand forecast is equated to the demand resolved at the previous time period. In the ROF and RCF algorithms, uncertainties in customer demand and process yield are considered and characterized by 50 independent scenarios having an equal probability of occurrence. The SCO problem is modeled with AMPL and solved using CPLEX 12.5 to a 1% optimality gap. Simulations were performed on a 3.4 GHz Intel® CoreTM i7 machine with 8 GB of RAM, running Windows 7 Professional 64-bit. Table 5 summarizes the model size of each optimization framework implemented.

Table 4 Summary of results with RCF, ROF and NOF for different uncertainty characterizations. Framework

NOF

ROF

RCF

Case

D

Y

B

D

Y

B

D

Y

B

J1 J2 × 10−4 FR

753 0.780 0.479

61.7 0.580 0.832

819 0.784 0.472

41.2 1.106 0.953

0.01 0.800 1.000

7.34 1.372 0.987

34.3 1.099 0.957

1.09 0.652 0.990

51.8 1.119 0.948

Note: Results correspond to the average performance over 50 independent closed-loop simulations D, uncertain demand; Y, uncertain yield; B, both uncertain demand and yield.

R. Mastragostino et al. / Computers and Chemical Engineering 62 (2014) 37–55

51

Fig. 9. Closed-loop result with NOF – Case Study II.

5.2.2. Comparing closed-loop performance with RCF, ROF, and NOF 5.2.2.1. Graphical comparison for single outcome. Figs. 9–11 show the performance of closed-loop simulations with the RCF, ROF, and NOF control approaches for the same outcome of demand and yield uncertainties. Due to space limitations, only representative trajectories are included in the results. All the cases are run with a  value of 100. Fig. 9 illustrates the closed-loop result with the NOF. A large amount of product back orders occurs for the product E after day 15 as shown by the back order trajectories, B (DC1 ), in Fig. 9. Inventory of final product E at both distribution sites is driven to zero and thus results in insufficient stock to fulfill customer demand.

The back order amount for the product G is comparatively lower than product E; however the inventory quickly vanishes to zero and therefore a back order situation arises after day 12. In the plots, the terms in parentheses indicate one or a combination of chemical, distribution center, manufacturing facility, production scheme, and demand. For example, PI (A-PS1 -M1 ) in the (2,2) subplot in Fig. 9 represents the consumption amount of material A in production scheme PS1 at plant site M1 . Fig. 10 shows the closed-loop result for the ROF approach. As it considers uncertainty information at each controller execution, adequate safety stocks for all materials are maintained throughout the supply chain system to meet varying demand. We can

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R. Mastragostino et al. / Computers and Chemical Engineering 62 (2014) 37–55

Fig. 10. Closed-loop result with ROF – Case Study II.

observe from Fig. 10 that it does not create a stockout condition for either product throughout the simulation run, shown by the trajectory of back orders (B), with the ROF, as compared to the NOF. Fig. 11 illustrates the closed-loop result with the RCF. The results closely match those of Case Study I. The operating cost for the RCF is 8.5% less than the ROF approach with same customer satisfaction level. Analogous to the ROF approach, it chooses to maintain safety stock to hedge against uncertainty and therefore it does not show any back orders for the entire simulation run. It should be noted that for certain demand uncertainty realizations, the NOF simulation encounters a problem of insufficient storage capacity as it does not consider the demand uncertainty and therefore tries to

store material in the storage if it is advantageous to do so. Therefore, the storage capacity constraints in the NOF simulations are implemented as soft constraints with a large penalty. 5.2.2.2. Comparison between multiple simulations. In this section, we compare the performance of the RCF and ROF control approaches with the NOF approach. The performance is reported with the average values obtained by running 20 independent simulation runs with a  value of 100. The simulation results are summarized in Table 6 and they show similar trends as in Case Study I. It can be seen from the table that the amount of back orders for the RCF and ROF approach is relatively low compared to NOF approach, which indicates a large improvement in customer service

R. Mastragostino et al. / Computers and Chemical Engineering 62 (2014) 37–55

53

Fig. 11. Closed-loop result with RCF – Case Study II.

Table 6 Summary of results with RCF, ROF, and NOF for different uncertainty characterizations - Case Study II Framework Case () J1 J2 × 10−4 FR CPU (s)

NOF

ROF

B (100) 948.9 5.619 0.665 0.14

B (100) 8.3 6.972 0.992 6.8

RCF B (100) 38.9 6.292 0.980 73.7

Note: Results correspond to the average performance over 20 independent closedloop simulations. B, both uncertain demand and yield.

level (J1 ). The average fill rate with the NOF is 0.65 as compared to 0.98 with the RCF and 0.992 with the ROF. Similar to Case Study I, the improvement in customer service for RCF and ROF approaches are achieved at the expense of higher operating cost. The ROF and RCF approach maintained sufficient levels of safety stock of all materials in the supply chain system to hedge against the demand and yield uncertainty and therefore they give rise to a higher operating cost. However, the increase in operating cost is lower in comparison to a sharp improvement in the customer service level. As expected, RCF outperforms ROF in terms of the operating cost. RCF achieved similar level of customer satisfaction as ROF at lesser cost (10.82%) as it considered predicted feedback information in the future time periods which helps RCF not to take overly conservative actions. The average computation

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time for controller execution (see Table 6) follows the same trend as in Case Study 1. The average computation time for the RCF and ROF is larger than the NOF but it still very modest considering the SCM sampling time of one day.

6. Conclusions and future work

Table A.3 Production cost parameter. Cost of PI in Cu

Plant sites

Production Schemes

M1 M2

In this work, we present a robust decision making tool for SCM, which addresses uncertainty in demand and model parameters explicitly. Through closed-loop simulation, the robust formulation is shown to substantially reduce the occurrence of back orders, as compared to a nominal MPC formulation, by maintaining a sufficient level of safety stock within inventory echelons. Of significance is that the robust framework maintains an appropriate level of safety stock dependent on the uncertainty characterization, which is a better technique than fixing safety stock levels on the basis of past data and experience. In this paper, we consider both open-loop and approximate closed-loop predictions of uncertainty propagation; the latter to mitigate the overly conservative nature of the open-loop approach to robust MPC. Simulation results provide favorable evidence to suggest that the approximate robust closedloop formulation provides an equivalent level of customer service at a reduced operating cost. Furthermore, this approach provides performance which is significantly less sensitive to the objective function weighting. This paper has applied the proposed framework to a networked supply chain system with multi-product plants having multiple production schemes. Future plans are to integrate decisions occurring at different time intervals, since in many cases production scheduling occurs at a finer resolution than transportation and purchasing. Further avenues for investigation include alternative strategies to solve multi-objective optimization problems, as well as decomposition and scenario reduction techniques to efficiently solve large-scale stochastic MILP problems.

Cost of PF in Cu

PS1

PS3

PS2

PS4

1.25 –

1 –

– 1.3

– 1.5

Table A.4 Shipment cost parameter. Plant sites

M1 M2

Cost of FP in Cu

Cost of FF in Cu

Plant sites

Distribution sites

M1

M2

D1

D2

– 4

4 –

2.7 2.8

2.8 2.5

Note: The shipment costs are the same for all chemicals within each shipment category. Table A.5 Inventory cost parameter. Parameter

Plant sites

Distribution sites

Cx

M1

M2

D1

D2

Cost of IR Cost of II Cost of IF Cost of IS

0.8 0.9 1.4 –

0.7 1.1 1.1 –

– – – 1.5

– – – 1.25

Note: The inventory costs are the same for each chemical within each inventory category. Table A.6 Transportation delays. ıR

Plant sites

ıS

Suppliers

Acknowledgments Financial support for this research through the Natural Sciences and Engineering Research Council of Canada (NSERC) Strategic Network on Value Chain Optimization (VCO) and Ontario Graduate Scholarship Program (OGS) is gratefully acknowledged.

M1 M2

Distribution sites

Plant sites

LS1

LS2

D1

D2

M1

M2

3 2

3 2

3 4

4 2

0 2

2 0

Table A.7 Production delays. Parameter

Appendix A. Case Study II data

ıP

Production schemes

M

ı

PS1

PS2

PS3

PS4

1

2

2

1

See Tables A.1–A.11 . Table A.8 Production yield parameter. Parameter

Table A.1 Simulation parameter values. Parameter

Value

Parameter

Value

n (days)  2 of a Range of DF

15 0.1 9 0–50

No. of scenarios Simulation length (days) Range of yield parameters T (days)

50 40 0.45–0.95 1

P  of ˇPS P 2 of ˇPS

Raw material

A B

M

LS2

1 1.4

1.2 1.7

PS2

PS3

PS4

0.8 0.0025

0.7 0.0025

0.8 0.0025

0.7 0.0025



Plant sites

Production schemes PS1

PS3

PS2

PS4

M1 M2

120 –

300 –

– 120

– 150

M1 M2

25 –

60 –

– 25

– 25

u

Suppliers LS1

PS1

Table A.9 Production batch size. Parameter

Table A.2 Raw material purchase cost.

Production schemes

Ml

R. Mastragostino et al. / Computers and Chemical Engineering 62 (2014) 37–55 Table A.10 Maximum transportation quantity. R

Plant sites

S

Suppliers

M1 M2

P

Distribution sites

Plant sites

LS1

LS2

D1

D2

M1

M2

120 120

120 120

100 100

100 100

0 100

100 0

Table A.11 Storage capacity. Parameter

˝R ˝I ˝EF ˝GF ˝ES ˝ES

Plant sites M1

M2

500 500 500 500 500 300

500 500 500 500 500 300

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