Preprints of the 18th IFAC World Congress Milano (Italy) August 28 - September 2, 2011
ON APPLICABILITY OF OPTIMAL CONTROL THEORY TO ADAPTIVE SUPPLY CHAIN PLANNING AND SCHEDULING
Dmitry Ivanov*, Alexandre Dolgui**, Boris Sokolov***
*University of Hamburg, Department of Business Administration Chair of Operations Management, 20146 Hamburg, Germany Phone: +49 371 53138947; E-Mail: [email protected] **Ecole Nationale Supérieure des Mines de Saint-Etienne Laboratoire en Sciences et Technologies de l'Information (LSTI) 158, Cours Fauriel, 42023 Saint-Etienne cedex 2, France E-Mail: [email protected] ***Insitute for Informatics and Automation of the RAS (SPIIRAS) V.O. 14 line, 39 199178 St. Petersburg, Russia; E-Mail: [email protected]
Abstract: Decisions in supply chain (SC) planning and scheduling are interconnected and depend a great deal on tackling uncertainty and dynamics of structures and processes in SCs that evolve over time. In this paper, we investigate the applicability of optimal control theory (OCT) to SC planning and scheduling based on the analysis of different streams in application of control theory to SCM and our own elabora tions. Some drawbacks and missing links in the literature are pointed out. Several crucial application areas of control theory to SCM are discussed. We conclude that with the help of control theory, stability, adap tability and disaster tolerance of SCs can be investigated in their fullness and consistency with operations planning and execution control within a conceptually and mathematically integrated framework. However, although SCs resemble control systems, they have some peculiarities which do not allow a direct applica tion of control theoretic methods. The combined application of OCT and operations research enriches the possibilities to develop solutions for many practical problems of SC management (SCM). At the same time, mathematics of OCT requires domain specific modifications to be consistent with discrete processes and decision making in SCM. We argue for a co operation between control experts and SC managers that has the potential to introduce more realism to the dynamic planning and models and improve SCM poli cies. Copyright © 2011 IFAC Keywords: supply chain; dynamics; planning; scheduling; control; optimal program control; adaptation; robustness.
1. INTRODUCTION faces the challenges of governing SC dynamics (Lee 2004, Graves and Willems 2005, Kouvelis et al. 2006). The term “ supply chain management ” (SCM) was coined in the 1980 90s. A supply chain (SC) is a network of organiza The research focus is now shifting to a paradigm that the per tions, flows, and processes wherein suppliers, cooperate and formance of SCs is to interrelate to dynamics, adaptability, coordinate along the entire value chain to acquire raw materi stability, and crisis resistance Stable SC processes in a com als, to convert these raw materials into specified final prod plex environment support enterprise competitiveness. On the ucts, and to deliver them to customers. contrary, the “overheated” SCs lack of resilience and stability (recent world financial crisis, natural catastrophes, and eve SCM studies human decisions on cross enterprise collabora ryday discrepancies in matching demand and supply evidence tion and coordination processes to transform and use the SC enough for it). resources in the most rational way along the entire value chain, from raw material suppliers up to customers, based on In these settings and in view of available IT, advanced inves functional and structural integration, cooperation, and coor tigations into SC dynamics and establishing adaptive feed dination. The impact of SCM on the changes in enterprise backs in SCs are becoming one of the major challenges in management paradigms can be compared with the develop SCM (Perea et al. 2000, Disney and Towill 2002, Braun et al. ments of total quality management (TQM) in 60 70s and 2003, Daganzo, 2004, Disney et al. 2006, Son and Venkates computer integrated manufacturing (CIM) in 80 90s. waran 2007, Chauhan et al. 2007, Meepetchdee and Shah 2007, Lee and Oezer 2008, Sarimveis et al. 2008, Ivanov and Along with considerable advancements in (optimal) SC de Sokolov 2010, Dolgui and Proth 2010b, Bartholdi et al. sign, planning and scheduling (Simchi Levi et al. 2004, Chen 2010). Indeed, an important part of SCM issues is concerned 2010, Hall and Liu 2011), the SCM research community with SC dynamics. Let us list a few of them. First, the issues
Copyright by the 423 International Federation of Automatic Control (IFAC) Preprints of the 18th IFAC World Congress Milano (Italy) August 28 - September 2, 2011 of performance and uncertainty regarding balancing effi In Table 1, we summarize possible applications of modern ciency, complexity, flexibility and robustness are to be CT results to SCM domain. named (Stevenson and Spring 2007, Meepetchdee and Shah Table 1 Applications of modern CT to SCM 2007, Wadhva et al. 2008). The main results of Application to SC control Second, the problem of “modelled” optimality and real life CT executability and adaptability is under consideration (Kra Criteria for existence Model verification for SC con jewski et al. 2005, Chauchan et al. 2007). Third, the problem of a solution trol atic of sustainability regarding balancing economic output and environment friendly resource and energy consumption Criteria for controlla Control processes verification is of great importance (Jaraman et al. 2007). In observing bility and attainability for a given time interval / De these problems it becomes evident that an important part of termination of the constraints re SCM problems is related to changes in the SC environment stricting SC control goal abilities and reacting to these changes, that is, with SC dynamics. Criteria for uniqueness Analysis of possibility to obtain of optimal program an optimal plan for SC control Another challenge of modern SCM is the SCs with both con control tinuous and discrete processes. Such SCs are typical in oil Necessary and suffi Preliminary analysis of optimal and gas industry, chemistry, etc. (Dessouky et al. 1999, Shah cient conditions of op program controls; generation of 2005, Kannegiesser et al. 2007, Puiginaer et al. 2008). In timality basic SC planning algorithms optimizing the performance of these SCs, the methods are required to consider both continuous and discrete processes. The program control SC planning, scheduling and ex In addition, even the SCs with only discrete processes fre and feedback control ecution control models on united quently contain different technological feedback flows, re methodological basis manufacturing processes, etc. along their product lifecycle Criteria for stability Evaluation of SC robustness and (Guide and Wassenhove 2009). and sensitivity sensitivity for environmental im pacts and for alteration of input The achievement of the planned SC performance can be in data habited by changes and perturbation impacts in a real execu tion environment (Kleindorfer and Saad 2005, Hendricks and In this study, we analyse the applicability of CT approaches Singhal 2005). Therefore, SCs are to be reliable and flexible to the SCM domain based on recent literature on dynamics in enough to be able to adapt their behaviour in the case of per SCM, recent literature on applications of control theory to turbations impacts in order to remain stable and resilient by SCM, and our own elaborations. Special attention will be recovering disruptions once disturbed. paid to optimal program control (OPC) and the domain of adaptive SC planning and scheduling. In these settings, the extensive development of approaches and models to tackling SC dynamics and considering SC The purpose of this paper is to describe the important issues planning and scheduling in terms of execution dynamics, and perspectives that delineate dynamics and adaptation in adaptation, and robustness is becoming a timely and crucial adaptive SC planning and scheduling, comment on methodi topic in SCM. A possibility to address the above mentioned cal issues, and describe in specific context of OPC possible challenges opens control theoretic approach. methods, models and algorithms in the adaptive SC planning and scheduling area. Control theory (CT) contains a rigor quantitative basis for planning optimal control policies including differential games The rest of this paper is organized as follows. We start with a and stochastic systems, stability of controlled processes and state of the art analysis. Section 2 analyses particular features non linear systems, controllability and observability, and of SCM problem regarding SC dynamics. Section 3 reveals adaptation (Pontrayagin et al. 1964, Lee and Markus 1967, general advantages and shortcomings of CT as applied to the Bellman 1972, Bryson and Ho 1975, Fleming and Richel SCM domain. In Section 4, we focus our discussion on OPC 1975, Casti 1979, Siliak 1990, Perea et al. 2000, Leigh 2004, and its application to SCM. In Section 5, we describe an Camacho and Bordons 2004, Bubnicki 2005, Lalwani et al. OPC based framework of interlinking SC synthesis and 2006, Disney et al. 2006, Sethi and Thompson 2006, Astrom analysis domains. We conclude the paper in Section 6 by and Wittenmark 2008, Sarimveis et al. 2008). describing the developed experimental environment and These tools can be applied for a wide range of systems, from summarizing the results of this study along with identifying discrete linear to stochastic non linear systems with both sta future research avenues in Section 7. ble and dynamically changing structures. CT can also be ap plied for analysis of equilibriums regarding resource con 2. ISSUES OF SUPPLY CHAIN DYNAMICS sumption and system output (Seierstad and Sydsaeter 1987, Sethi and Thompson 2006). These tools can be used regard Only a few years have passed since SCM has been consid ing SC sustainability analysis. Besides, optimal CT provides ered just as an extension of logistics or procurement man an extensive approach to optimal planning and scheduling of agement. Nowadays, the understanding of SCM as a wider both continuous and discrete processes (Hwang et al. 1967, concept and, actually, as an independent scientific discipline Kogan and Khmelnitsky 2000, Sethi and Thompson, 2006). and as one of the key management functions in enterprise is
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Preprints of the 18th IFAC World Congress Milano (Italy) August 28 - September 2, 2011 widely understood (Chen and Paulraj 2004, Christopher In OR, improvements in SC planning and scheduling are usu 2005, Harland et al. 2006, Chopra and Meindl 2007; Simchi ally algorithmic and refer to the methods of linear program Levi et al. 2003). ming, integer programming and dynamic programming (Sim chi Levi et al. 2004). However, high dimensions, dynamics, With the development of SCM, new specific problems and uncertainty, and complexity of real problems challenge the integrated problems from production and logistics manage optimization and frequently lead to application of heuristics ment such as distribution production or production inventory or combination of meta heuristic with mathematical pro problems have arisen (Simchi Levi et al. 2004, de Kok and gramming (MP). Increase in complexity and multi Graves 2004, Chen 2010). SCs are becoming more and more dimensionality of SCM problems necessitates that OR me complex and dimensionally larger. In order to conduct re thodology closes with systems and control theory, artificial search on such complex systems, a combination of different intelligence, and informatics. methods becomes necessary. Indeed, it is becoming more and more difficult to represent the ever more complex SCM prob Summarizing, problems of SC planning and scheduling are lems within one big planning model. challenged by high complexity, combination of continuous and discrete processes, integrated production and transporta SCs are characterized by a set of interrelated structures such tion operations as well as dynamics and resulting require as organizational, functional, informational, financial, etc. ments for adaptability and stability analysis. Therefore, SCs (Ivanov et al. 2010). Decisions in all the structures are inter are to be considered as adaptive systems. A possibility to related. Moreover, the structures are subject to changes; address the above named issues opens CT and optimal pro hence, SC structure dynamics (Okhtilev et al. 2006) is fre gram control (OPC) in particular. quently encountered. Furthermore, the SC elements are ac- tive . A SC is characterized by uncertain interactions of its CT is becoming of a greater interest to researchers and practi elements (enterprises and SC managers) and distributed goals tioners (Perea et al. 2000, Braun 2003, Daganzo, 2004, Dis and possible conflicts (Swaminathan 1998, Surana et al. ney et al. 2006, Lalwani et al. 2006, Hoberg et al. 2007, 2005, Kuehnle 2008). Goettlich 2007, Wang et al. 2007, Sarimveis et al. 2008, Iva nov 2009, Schwartz and Rivera 2010). Besides, the SC execution is accomplished by permanent changes in the internal network properties and the environ CT is favorable in the cases of many dynamically changing ment. The changes set limits on the SC performance . The control parameters, obtaining analytical solutions or proper limits on the performance require the stability and robustness ties, and in investigating different mutual impacts of SC analysis (Mesarovich and Takahara 1975) and establishing planning and scheduling parameters (e.g., demands, resource SC adaptation to a real environment. The dynamics, feed and channel capacities, lead time, lot sizes, and inventories) backs and not determined considerations of future make SC on the SC tactical and operative performance (i.e., service process non-stationary and non-linear (Narendra 2005). level and costs). In some cases (e.g., if many changes, many stages, and many periods), it is convenient to transit from a In addition, in recent years, the works on covering SC dy discrete problem statement to continuous solution procedure, namics have been extended by developments in information and then represent the result again in discrete terms due to technologies such as RFID (Radio Frequency Identification), particular accuracy of continuous time models. SCEM (Supply Chain Event Management) and mobile busi ness provide a constructive basis to incorporate the stages of Attraction of CT can be seen as the next crucial step in the SC planning and execution (Lee and Oezer 2008, Sokolov et development of SCM theory to reflect the real time dynamics al. 2010). In these settings, strategic design of SCs (SCD), and dynamic optimization of SC structures and processes as planning and scheduling of orders in SCs (SCP), and SC exe well as explore robustness, stability, and adaptability in the cution control (SCE) is to consider integrated on the basis of real time mode taking into account non linearity, non feedback adaptation principles. stationarity and uncertainty in SCs. Another challenge in SCM is the problem of dynamics and The purpose of this paper is to describe the important practi maximization of performance. In these settings, the duality of cal issues and perspectives for CT application to SCM, com the SCM goals – maximizing the service level and minimiz ment on methodical issues, and describe one specific context, ing costs – should be enhanced by the third component – model and algorithm in the dynamic SC planning and sched maintaining SC stability (Ivanov and Sokolov 2010). uling area. In discrete optimization of operations research (OR), remark able advancements have been achieved in SC planning and 3. PRO AND CONTRA OF CONTROL THEORETIC AP scheduling for the last decade for the domain of production– PROACHES TO SCM distribution models at strategic tactical (Vidal and Goet schalckx 1997, Chandra and Grabis 2007) and operative lev Dynamics in SCs can be referred to both the dynamics of a els (Hall and Potts 2003, Chen and Vairaktakaris 2005, Sark process under optimization (dynamics of the transition from er and Diponegero 2009, Chen 2010, Hall and Liu 2011). an input to an output state) and the real time dynamics re garding the feedback loop consideration and adaptation in Along with considerable advancements in discrete SC opti accordance to an actual execution environment. Attraction of mization, the domain of SC dynamics analysis and adaptive CT can be favorable for both the SC synthesis and analysis (re)planning and (re)scheduling still remains ill investigated.
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Preprints of the 18th IFAC World Congress Milano (Italy) August 28 - September 2, 2011 stages. nov’s stability or BIBO stability. First, the classical models imply natural movement of objects. Second, they typically Sarimveis et al. (2008) underline the resemblance of SCs to consider very small deviations of control and output va engineering dynamic systems. Previously, CT has been ex riables. Third, stability analysis can help in estimating SC tensively applied for management and economics applica volatility in any concrete state. But it is not enough to stabil tions (Seierstad and Sydsaeter 1987, Bubnicki 2005, Sethi ize the SC – the SC should also bring profits; hence, the in and Thompson 2006). In the SCM domain, application of CT clusion of performance considerations (i.e., the robustness has been increasingly developed since 1980s beginning with analysis) is required as the next step. Finally, classical stabili the works on inventory control (e.g., Axsaeter 1986) and SC ty analysis is concerned with funding equilibrium states for dynamics (e.g., Wikner et al. 1991). The applied tools vary mechanical and automatic systems. from classical transfer function analysis to model predictive control. In reviewing the literature, the following problem A very extensive area of CT applications to SCM domain is domains can be indicated: related to the adaptation and real time control (Perea et al. 2000, Braun 2003, Wang et al. 2007, Puigjaner and Lainez 1) Dynamic inventory control policies, 2008, Sarimveis et al. 2008, Schwartz and Rivera 2010). Ac 2) Analysis of disturbances and fluctuations (e.g., bullwhip tually, it is the area for many SC scholars and professionals to effect, stability, and robustness), which they refer first while talking about the CT. 3) Adaptation and real time control, and Adaptive control (AC) is a control strategy with some form of recursive system identification (Astrom and Wittenmark 4) Optimal multi stage, multi period production planning. 2008). However, classical AC approach has not found a wide application to the SCM domain so far. Let us consider ecent literature in these domains. A popular technique of SC adaptation in the modern CT is Dynamic inventory control policies have been previously the model predictive control (MPC). In MPC, a system model considered in studies by Axsaeter (1985), Axsäter and Rosl and current and historical measurements of the process are ing (1993), Grubbström and Wikner (1996), Ortega and Lin used to predict the system behaviour at future time instants. A (2004). Disney et al. (2006) and Hoberg et al. (2007) investi control relevant objective function is then optimized to calcu gated recently the effects of inventory control policies on late a control sequence that must satisfy the system con order and inventory variability with linear classical CT. straints. The MPC approach is not simply to run the planning Bensoussan et al. (2007) considered possible information frequently, but rather to develop decision policies (Wang et delay and incompleteness in the ordering policies for invento al. 2007). Applications of MPC to multi echelon production– ry decisions. However, the authors frequently point out cer inventory problems and SCs have been examined previously tain analytics limitations. For further reading on application in the literature. Perea et al. (2000) modeled multi plant, mul of classical CT to SCM, we refer the readers to the study by ti product polymer process through difference equations and Sarimveis et al. (2008). schedule optimisation in MPC framework. Braun et al. (2003) developed a decentralized MPC implementation for a Another fundamental domain of CT is related to analysis of six node, two product, and three echelon demand network disturbances, fluctuations, robustness and stability of SCs. problem. In the study by Puigjaner et al. (2008), a multi stage The studies by Daganzo (2004) and Warburton et al. (2004) stochastic model has been employed. However, the stabiliz applied classical control theory and Lyapunov’s stability me ing controllers still remain a critical bottleneck in MPC trics to the SCM domain. Disney and Towill (2002) applied a (Mayne et al. 2000). discrete linear control theory model to determine the dynamic stability of vendor managed inventory (VMI) SCs. Warbur A critical issue in applying MPC to SCM is the centralized ton et al. (2004) provided a stability boundary for the conti controller and its functions. In technical systems, the control nuous time SC ordering decision with regard to BIBO ler is a technical device (e.g., a sensor) that adapts the system (bounded in bounded out) stability. behavior based on error identification within milliseconds. The controller in the SC is a man ager, or more precisely, a In some studies, bullwhip effect has been addressed from number of managers with possible interest conflicts. Even if a control theoretic perspective. A sys tem control framework deviation in the SC execution has been identified (e.g., delay was recently introduced to study the bullwhip effect (Dagan identification with the help of RFID), the MPC controller will zo 2004; Dejonckheere et al., 2004). Variance formulas for not be able to change anything in this situation. The role of the orders by any supplier have been derived for multi stage this model will be to identify the deviations, to notify the SC serial chains and any ergodic customer demand (Ouyang managers, to estimate the impact of this disturbance on SC 2007). Various strategies to avoid or mitigate the bullwhip performance, and to develop any recommendations on the have been discussed by Disney et al. (2006) and Quyang adaptation. That is why additional research is needed for ana (2010) who analyzed the bullwhip effect in multi stage SCs lyzing the applicability of MPC to human driven SC adapta operated with linear and time invariant inventory policies and tion. shared SC information. The last application domain of CT to SCM is optimal multi Although stability analysis is an useful tool, it is subject to stage, multi period planning and scheduling. Let us discuss many restrictions if applied in the classical form of Lyapu this domain in details in next section.
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4. OPTIMAL PROGRAM CONTROL FOR ADAPTIVE The stream of production scheduling has been continued by SUPPLY CHAIN PLANNING AND SCHEDULING Kinemia and Gershwin (1983), who applied hierarchical me thod in designing the solution procedure to the overall model, 4.1 Recent applications and Khmelnitsky et al. (1997), Kogan and Khmelnitsky (2000), who applied the maximum principle in discrete form Optimal program control (OPC) is a method for solving dy to planning continuous time flows in flexible manufacturing namic optimization problems, when those problems are ex systems and transited from the hierarchical approach to heu pressed in continuous time and the value of a goal criterion ristic rules for OPC calculation. (or a number of criteria) are accumulated over time. OPC is a deterministic control method as opposite to the stochastic The developments of this period of time have been docu optimal control (Fleming and Richel 1975). One of the basic mented in surveys by Hartl et al. (1995) and Sethi and milestones in modern OPC, along with dynamic program Thompson (2000). These studies have also extensively consi ming, is the maximum principle that was developed in 1950s dered the domain of coordinated production marketing strat by Russian mathematicians among whom the central charac egies regarding simultaneous manufacturing and price deci ter was Lev Semenovich Pontryagin. sions. The work in this domain has started with the study by Pekelman (1974) and has been continued by Sethi’s advertis Maximum principle is an original method for computing op ing model (Sethi and Thompson 1981), optimal pricing and timal control (OC) when optimizing system behavior over production model by Feichtinger and Hartl (1985), and many many periods of time under constrained control subject to other applications up to now (e.g., Gaimon 1988, Abad 1989, several decision variables where other techniques can become Feichtinger et al. 1994, Kogan and Herbon 2008, He et al. analytically and computationally difficult to apply. The initial 2009). In recent years, continuous time maximum principle formulation of maximum principle was concerned with the has been applied to revenue management and dynamic pric problem of transfer a space vehicle from one orbit to another ing as a stochastic optimal control problem (Levin et al. with minimum time and minimum fuel consumption. 2008). According to the maximum principle, the optimal solution of Although the OCT application has been widely understood at the instantaneous problems can be shown to give the optimal the tactical planning level, the research on OCT for detailed solution to the overall problem (Pontryagin et al. 1964, Bol dynamic production and transportation scheduling in the in taynskiy 1973, Sethi and Thompson, 2006). Maximum prin tegrated SC context is fairly recent, although there is a wealth ciple basically generalizes the calculus of variations and of works in these direction from the enterprise perspective builds the basis of the modern OCT. Even the development and treatment those problems in isolation (production sche of the maximum principle for optimal dynamic system con duling and routing). This rapidly emerging field of integrated, trol with constrained control variables has stimulated the ap customer oriented SC scheduling (Chen 2010) can become a plication of OCT to industry and engineering. new application area for OCT. OCT has been extensively applied to production and logistics 4.2 Practical needs for application of optimal control and planning and scheduling both for continuous and discrete maximum principle to SC optimization systems right from the beginning. In particular, the following problem domains have been addressed with the help of max Recall that according to the maximum principle, the optimal imum principle in production and logistics areas so far: solution of the instantaneous problems can be shown to give the optimal solution to the overall problem. If so, it is a very 1) multi level, multi period master production scheduling convenient approach to naturally decompose a problem dy problems with lot size and capacity optimization, namics horizontally into some subproblems to which optimal 2) coordinated production marketing strategies regarding solutions can be found, e.g., with the help of mathematical simultaneous manufacturing and price decisions, programming, and then link these solutions with the help of OPC. 3) inventory control, and This property is of a great practical importance for SC opti 4) revenue management and dynamic pricing. mization. Indeed, it is frequently difficult or impossible to accumulate all the necessary information on SC dynamics at Beginning with the study by Holt et al. (1960) that seems to the initial planning t point of time. In this setting, adaptive be the first study on using calculus of variations to solve pro 0 planning and scheduling concepts are frequently applied duction inventory problems, the work by Fan and Hwang when a plan is modified periodically by change in the SC (1964) and Hwang et al. (1967) were among first studies on parameters or the characteristics of control influences on the application of discrete maximum principle to multi level and basis of information feedback about a current system state, multi period master production scheduling and inventory the past and the updated forecasts of the future (Ivanov and control. Hwang et al. (1967) determined production planning Sokolov 2010). as optimal control action and corresponding trajectory of state variables (the inventory) with the maximum principle Another practical challenge is flexible resource, capacity, and subject to minimization of costs. First overviews of the OCT flow allocation to dynamically changing environmental and for economics and management domain have been developed internal conditions (e.g., demand, SC structure, collaboration by Bensoussan et al. (1974) andSethi and Thompson (1981). and coordination rules). Integrated logistics planning by 4PL
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(fourth party logistics) service providers face along with a with changing structural characteristics has been developed. detailed treatment of dynamic parameters such as varying This idea is based on the observation that that during the capacities in problems with multiple plants and distribution planning horizon, different structural elements (decision centres at different locations. An increasing number of com makers, processes, products, control variables, constraints, panies now adopt make to order and assemble to order con goals, perturbations, etc.) are involved in decision making on cepts. In many industries (e.g., perishable or seasonal prod SC planning, and not all of them at the same time. In moving ucts or process industry), finished orders are frequently deli on through the planning period, these elements appear and vered to customers immediately or shortly after production disappear from the decision making. If so, there is no need to without intermediate inventory. In general, specific SC colla consider all the structural elements at the same time in a large boration and coordination evidence to extend the models of planning problem in steady state environments. Moreover, SC optimization to the dynamics domain. the solution procedure becomes undependable from the con tinuous optimization and can be of discrete nature, e.g., a Last but not least – a crucial topic is the impact of uncertainty linear programming, transportation problem, or integer allo and disruptions. Big centralized models for planning the cation problem (Ivanov and Sokolov 2011). whole time horizon are very sensitive to changes in data availability. The existence of a great diversity of different This idea is to some extent similar to those in combining MP dynamic characteristics in those problems SCs can signifi and meta heuristics that uses an exact method over restricted cantly impact SC performance. SCM is based on information portions of the solution space subject to a given problem of a sharing and coordination, and many SC optimization model very large feasible space. By taking optimal decisions within assume full information availability. However, due to dynam these certain intervals, we can address the problems of signif ic changes and coordination problems in the SC it is frequent icantly smaller dimensionality. This means, that the set of ly impossible. If such a disturbance takes place two issues feasible solutions is presented dynamically, but the solution occur: Is the SC able to continue its operation? Can mathe at each point of time are calculated at the local section and matical models work with incomplete or delayed informa for deterministic problems very small dimensionality. This is tion? In the light of the above mentioned practical challenges, very important as the computational time decrease considera the application of OPC and maximum principle to SC opti bly even if a large number of nodes or arcs area considered mization can be very favorable. and additional constraints are introduced. Besides, the a priori knowledge of the SC structure, and moreover, structure dy 4.3. Methodical challenges for application of OPC to SC namics, is no more necessary. optimization Let us present a possible scheme for applying maximum The application of the OPC to SC optimization is not a trivial principle to an SC scheduling problem that is similar to job problem. Discrete time and discrete quantities of SC opera shop scheduling (Ivanov and Sokolov 2011). The scheduling tions in both production and logistics SC parts can make the model is formulated as a linear non stationary finite SC planning and scheduling problems intractable. Despite dimensional controlled differential system with the convex OC models make it possible to reflect dynamics, the consid area of admissible control. The non linearity is transferred to eration of sequencing and resource allocation in these models the model constraints. This allows us to ensure convexity and is significantly complicated by mathematical features. to use interval constraints. Besides this, the required consis For example, the derived function from the arising sectionally tency between OPC and MP models is ensured – although the continuous functions (Moiseev 1974) is infinity. In addition, solver works in the space of piecewise continuous functions, such problems as numerical instability, nonexistence of gra the control actions can be presented in the discrete form as in dients, and non convexity of state space should be named. In MP model. addition, the problem of continuous time and state variables The developed model formulation satisfies the conditions of in canonical OPC statements and discrete times and quanti the existence theorem by Lee and Markus which allows us to ties in SCs exist. SC scheduling could not be performed in assert the existence of the optimal solution in the appropriate applying conventional form of OPC formulation. class of admissible controls and to calculate the OC with the However, the maximum principle permits the decoupling of help of maximum principle. This is the essential structural the dynamic problem over time using what are known as ad property of the proposed approach in order to apply discrete joint variables or shadow prices into a series of problems optimization for OPC calculation. In maximizing Hamilto each of which holds at a single instant of time. This property nian in OC computing, this makes it possible to solve the of optimal control is very helpful when interconnecting MP assignment problem and the flow distribution problem both and OPC elements. in discrete and continuous manner. In this aspect, the pro posed approach differs from the scheduling with the help of maximum principle with only continuous control variables or 5. OPC BASED ADAPTIVE SUPPLY CHAIN PLANNING discrete maximum principle. The model can work with both AND SCHEDULING continuous and discrete processes. The discretization is poss ible since the optimization problem is in fact a LP/IP prob 5.1. Synthesis (planning) domain lem. This is mainly due to the fact that the governing dynam ics in the supply network are linear in the state (but not in the In the studies by Ivanov et al. (2010) and Ivanov and Sokolov control) variables. (2010), an original SC representation as a dynamic system
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On the basis of the maximum principle, the original problem pose to apply the method of attainable sets (AS) to calculate of OPC is transformed to the boundary problem. Then a re the robustness index for SC schedules and to obtain the at laxed problem is solved to receive OPC vector (i.e., the SC tainable sets for interval data with no a priori information schedule). For OC computation, the main and conjunctive about perturbation impacts, i.e., for the most severe case of systems are integrated. The control vector at time t = t0 re non stationary perturbations. turns a maximal value to the Hamiltonian. Then we make the The AS approach is to determine a range of operating poli first integration step with the value of control vector at time t0 and again implement the maximum principle to receive the cies (the union of which is called as an AS) during the sche duling stage over which the system current performance can next value for time t=t1. The process of integration is contin ued until the end conditions are satisfied and convergence be guaranteed to meet certain targets, i.e., the output perfor accuracy is adequate. For obtaining the vector of conjunctive mance (Ivanov and Sokolov 2010). The AS characterizes all equation system, the Krylov Chernousko method is used that possible states of the SC schedule subject to different varia is based on joint use of modified successive approximations tions of SC parameters in nodes and channels (e.g., resource method and branch and bound method. capacity availability). Let us turn to the adaptation level. At the scheduling stage, The AS is calculated from the main OPC vector. Here, the different levers (material inventory, financial reserves, and perturbation impacts play the role of control variables. In IT) to mitigate uncertainty and to ensure SC execution con varying these perturbations at each instant of time over the trol under the presence of disturbances are built. As the SC schedule within the time interval and setting these variations execution is inevitably followed by changes of both environ into the initial differential system, a set of points where the ment and SC itself, the adjustment of SCs is needed. A con SC schedule can be steered to is generated. In other words, a venient way to approach this issue is the concept of adapta set of alternative OPC vectors is generated through admissi tion. ble variations of perturbation impacts and forms herewith the AS of the SC schedule under disturbances. The main purpose of the adaptation framework is to ensure SC tuning with regard to changes in the execution environ However, if the dimensionality of control and state vectors is ment and planned values of performance indicators. In Fig. 1, high, the construction of an AS is a rather complicated prob the adaptive framework is presented (based on Skurikhin et lem. That is why an AS is usually approximated in different al., 1989). forms. E.g., the construction of AS can be restricted to a sim ple (rectangular) form through the four point orthogonal pro External 3 jection of the convex attainable sets in the state and perfor adaptation mance spaces on each other.
Plan 4 Scheduling 5 6 Analysis With the help of AS, it becomes possible to create the dy formation model namic projection of the SC schedule execution on different 2 Analysis uncertainty scenarios. Therefore, AS can be used to calculate 7 8 Internal Plan adaptation execution the corresponding robustness metric for SC schedules. This metric can be used for ranging alternative SC schedules and Perturbation corresponding SC plans subject to individual risk perceptions 1 9 influences Monitoring Process of decision makers.
Figure 1. SC adaptive scheduling and execution control With the help of AS, scheduling decisions can be brought into correspondence to the higher level decisions and used as A hierarchy of adjustment actions is brought into correspon estimates for input data needed for making planning deci dence with different deviations in the SC execution (Ivanov sions. The schedule can be analysed with regard to both out and Sokolov 2010). At the constructive level, the above put performance indicators and robustness. If none of the mentioned integration is realized by the state variables and generated schedules provides a satisfactory level of perfor conjunctive variables that memorize the schedule execution mance and robustness, parameters of the SC plans (e.g., re and use this information for schedule update using the maxi source capacities, inventories, lot sizes, delivery data, etc.) mum principle. can be tuned. Besides, such an analysis can reveal that a very cost intensive SC plan attains the same schedule robustness 5.2. Analysis domain as a more cost efficient SC plan. An SC planner can analyse A crucial application area of CT to SCM is SC dynamics the perturbation impact on SC schedules and how these analysis. Even in this area, the potential of CT can be applied changes influence the planning output performance indica to SCM to great extent. For example, it becomes possible to tors. investigate the SC robustness as an ability to continue sche dule execution and to achieve the planned output perfor 6. EXPERIMENTAL ENVIRONMENT mance in the presence of disturbances. Continuous optimization is a challenging calculation task. With an original representation of the SC schedule as OPC, Thus any sensible judgments on the models and algorithms robustness objective can be integrated as a non stationary can be made only by application of special tools. For the ex performance indicator in SC scheduling decisions. We pro periments, we elaborated the models in a software package.
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The software has three modes of operation with regard to tors in dynamics that are difficult to express within a static scheduling and an additional mode to analyse robustness of and discrete time models. the schedules. The first mode includes the interactive genera tion/preparation of the input data. The second mode lies in Third, the possibility of covering the SC dynamics at the the evaluation of heuristic and optimal SC schedules. The process level and the changes in SC and environment is also a following operations can be executed in an interactive re strong contribution of CT. Fourth, CT allows the considera gime: tion of goal oriented formation of SC structures and the solu tion of problems in this system as a whole. Fifth, as it has • multi criteria rating, analysis, and the selection of SC plans been proved in our research, OPC can applicable for SC and schedules; planning and scheduling with both discrete and continuous processes. In using CT, important categories of SC analysis • the evaluation of the influence that is exerted by time, eco such as stability, robustness and adaptability can be taken nomic, technical, and technological constraints upon SC into consideration. In addition, the dimensionality of OR structure dynamics control; and based problems can be relieved with the help of distributing • the evaluation of a general quality measure for SC plans model elements between an OR based (static aspects) and a and schedules, and the evaluation of particular performance CT based (dynamic aspects) model. indicators. At the same time, mathematics of CT requires domain The third mode provides interactive selection and visualiza specific modifications to be consistent discrete processes and tion of SC schedule and report generation. An end user can decision making in SCs. Therefore, the critical step in con select the modes of program run, set and display data via a trol theoretic formulating a SCM problem is the interpreta hierarchical menu. tion of SC material processes, variables, goals, and informa tion flows in terms of, e.g., optimal control. After that, the With regard to perturbation impacts, an SC planner can ana OC model itself should be formulated in such a way to allow lyse different alternative SC plans and schedules, fill these OC computation. plans with reliability and flexibility elements to different ex tents, and then analyse how these changes influence the key Although in certain case its application can lead to computa performance indicators. In the current version of the software tional problems, it is possible to formulate the OPC model in package, this tuning is still performed manually; however, the such a way to allow efficient OPC computation. The main extension of the software prototype in this direction is under idea of the proposed model is to implement and update (e.g., development. due to dynamic changes in capacity availability) non linear constraints in convex do main of allowable control inputs The experiments have been conducted with the help of self rather than in the right parts of differential equations. The programmed C++ algorithm that creates files to address Mat proposed substitution lets use fundamental scientific results Lab MP library while maximizing Hamiltonian. The con of the OC theory in various SCM problems (including sched ducted experiments showed that the application of the pre uling). sented dynamic scheduling model is especially useful for the problems where jobs are arranged in a certain order (e.g., Another challenge is to facilitate the discreteness of decision technological restrictions). This is the case in SC planning making in SCs while using continuous OC models along with and scheduling. The computational experiments have proved the ensuring a possibility to compute the solution in widely convergence and tractability of the proposed algorithm and available optimization software for OR. More efficient algo model. They have also shown, that the convergence depends rithms for computing OC are needed to make OPC a com mainly on the selection of the conjunctive system vector petitive tool for SC optimization. (here, we have used priority rules, however, in future, it can However, ipso facto that an SC plan and schedule can be be interested to investigate the application of meta formulated as OPC is a great advantage subject to further heuristics). Besides, the proposed method is especially useful dynamics analysis that is a crucial application area of CT to in the case of many conflicting resources and capacity defi SCM. Even in this area, the CT can be applied to SCM to cits. great extent and enlarge the scope of SC analysis that is cur rently rather limited. 7. CONCLUSIONS Finally, the developed model and algorithm have properties One of the fascinating features of CT is the extraordinary that are unique within OPC, e.g., taking into account logical wide range of its possible applications. The first strong con constraints, non interruption of jobs, and discrete time jobs tribution of CT to operations and SCM regarding the dynam along with continuous maximum principle application which ics is the interpretation of planning and execution processes allow application to both discrete and continuous SC proc not as isolated domains but as an adaptive process. esses. Second, an advantage of CT is the possibility of solving Basically, the developed approach to OPC for SC planning problems with non stationary and non linear processes due to and scheduling is applicable to the following problem do the independency of time variable. Continuous dynamic main: dynamic allocation of jobs with frequently changing models let us establish and optimize SC performance indica parameters to capacity-constrained limited resources subject to the best possible use of time and material flows.
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With the formulation of the dynamic scheduling model of CT with OR and artificial intelligence can potentially en within the proved fundamental theories of OCT, new specific rich the possibilities to develop useful solutions to many SC coordination and collaboration features and key perform practical problems. ance indicators (e.g., maximizing equal resource charge and service level for each order, flexible capacity and job alloca ACKNOWLEDGEMENTS tion, etc.) and implementing adaptive planning concept sub ject to interlinking planning and scheduling stages can be The research described in this paper is partially supported by implemented. grants from the Russian Foundation for Basic Research To the crucial practical directions for future research belong (grants 11 08 01016 а, 11 08 00767 а, 11 07 90411 Ukr а, (1) integrated logistics planning by 4PL providers in dynamic 09–07–00066 а, 10–07–00311 а, 10–08–90027 Bel а) and allocation problems with multiple plants and distribution cen Department of nanotechnologies and information technolo tres at different, (2) make to order and assemble to order gies of the RAS (project О 2.5/03). concepts, (3) SCs with perishable, seasonal products or proc ess industry) where finished orders are frequently delivered to customers immediately or shortly after production without REFERENCES intermediate inventory and balancing of production and transportation lot sizes and capacities becomes a crucial suc Abad, P.L. (1989) 'Multi product multi market model for co cess issue. ordination of marketing and production decisions', Inter national Journal of Systems Science, 20: 11, 2011 2027 Dynamics of capacity availability and flexible order coordi Astrom KE., Wittenmark B. (2008). Adaptive control (2nd nated order allocation in the SC are also practical challenges. edition). Dover. In general, more specific SC context (i.e., integration, col Axsäter, S. (1985) Control theory concepts in production and laboration and coordination) is needed to extend the existing inventory control. International Journal of Systems models of job delivery coordination to the SCM domain. Science. 16(2):161 169. Axsäter, S., Rosling, K. 1993. Installation vs. echelon stock Another task of future research is to compare centralized and policies for multi level inventory control. Management decentralized control strategies. Centralized OPC strategies Science 39, 1274 1280. may potentially generate better solutions rather than decen Bartholdi III JJ, D.D. Eisenstein, Y.F. Lim (2010) Self tralized control. At the same time, centralized SC control can organizing logistics systems, Annual Reviews in Control be only conditionally performed due to the decentralized de 34(1): 111 117. cision making nature in SCs. Here, differential game OPC Bellmann R (1972) Adaptive control processes: a guided based models can be investigated. tour. Princeton Univ. Press, Princeton, New Jersey. The incorporation of intelligent IT into CT, also known as Bensoussan, A., Çakanyildirim, M., & Sethi, S. 2007. Opti intellectualization of control , can provide a variety of meth mal Ordering Policies for Inventory Problems With Dy ods and tools for dynamics in the SC domain. This can be namic Information Delays. Production & Operations come the area where the knowledge of SC managers and con Management, 16(2), 241 256. trol specialists can be effectively integrated taking advantages Bensoussan, A., Hurst, E.G. and Näslund, B., 1974. Man of modern IT, e.g., for investigating SC dynamics or applying agement Applications of Modern Control Theory. , RFID to SC monitoring and adaptation. North Holland, Amsterdam. Boltyanskiy, B. (1973). Optimal control of discrete systems. We can conclude that the success of SC planning and control Moscow, Nauka. depend a great deal on tackling uncertainty and dynamics in Braun MW, Rivera DE, Flores ME, Carlyle WM, Kempf KG SCs that evolve over time. CT provides a variety of methods (2003) A model predictive control framework for robust and tools for the SCM domain and allows to take into ac management of multi product, multi echelon demand count dynamics, real dimensions, non linearity and non networks. Annu Rev Contr 27:229–245 stationary of SC processes. In addition, with the help of CT, Bryson, A.E., Ho, Y. C. (1975). Applied Optimal Control. non stationary performance indicators such as robustness and Hemisphere, Washington. stability can be investigated in their fullness and consistency Bubnicki Z. 2005. Modern control theory. Springer: Berlin, with operations planning and execution control within a con Heidelberg. ceptually and mathematically integrated framework. Camacho EF, Bordons C (2004) Model predictive control. Springer, London In future, an interdisciplinary collaboration between SCM Casti, J.L.: Connectivity, complexity and catastrophe in and control researchers is needed. Often, researchers in logis large scale systems. Wiley Interscience, New York and tics and SCM apply interesting CT techniques to solve exist London (1979) ing problems, but do not have close contact to the progress of Chandra,C., Grabis, J.: Supply chain configuration. Springer, research in this field. Similarly, CT researchers are often not New York (2007) aware of the possibilities to connect their work to state of the Chauhan, S.S, Gordon, V., Proth, J. M. (2007). Scheduling in art logistics and SCM. In limiting the decision making sup supply chain environment. European Journal of Opera port on only methods of CT, the domain of SC dynamics tional Research, 183(3), 961 970. would still remain ill investigated. The combined application
431
Preprints of the 18th IFAC World Congress Milano (Italy) August 28 - September 2, 2011
Chen Z. L. Integrated Production and Outbound Distribution Guide V.D.R. Jr., L.N. Van Wassenhove, “The evolution of Scheduling: Review and Extensions. Oper Res, Vol. 58, closed loop supply chain research”, Oper Res., Vol. No. 1, 2010, pp. 130–148 57(1), pp. 10 18 (2009). Chen, I.J., Paulraj, A. 2004 Understanding supply chain man Hall NG, Liu Z. (2011). Capacity allocation and Scheduling agement: Critical research and a theoretical framework in Supply Chains. Oper Res, to appear, DOI International Journal of Production Research 42 (1), pp. 10.1287/opre.1090.0806. 131 163 Hall, N. G., C. N. Potts. 2003. Supply chain scheduling: Chen, Z. L., G. L. Vairaktarakis. 2005. Integrated scheduling Batching and delivery. Oper. Res. 51 566–584. of production and distribution operations. Management Harland C. et al. (2006) Supply Management: Is It A Discip Sci. 51 614–628. line? Int. Journal of Operations and Production Man Chopra, S. and Meindl, P.: Supply chain management. Strat agement, 25(7), 730–753. egy, planning and operation. Pearson Prentice Hall, New Harrison TP (2005) Principles for the strategic design of sup Jersey (2007) ply chains. In: Harrison TP, Lee HL, Neale JJ (Ed) The Christopher M. Logistics and Supply Chain Management: practice of supply chain management. Kluwer Academic Creating Value Adding Networks. Financial Times, Publishers, Boston. Prentice Hall: Dorchester; 2005. Hartl, R.F., Sethi, S.P., Vickson, R. (1995), A survey of the Daganzo, C.F. On the stability of supply chains. Operations maximum principle for optimal control problems with Research 2004, Vol. 52, No. 6, pp. 909 921. state constraints. SIAM Review 37 (2), 181 218. Dejonckheere, J., Disney, S.M., Lambrecht, M.R., Towill, He, X.; Prasad, A.; Sethi, S.P. (2009). "Cooperative Advertis D.R., 2004. The impact of information enrichment on the ing and Pricing in a Stochastic Supply Chain: Feedback bullwhip effect in supply chains: A control engineering Stackelberg Strategies". Production and Operations perspective. European Journal of Operational Research Management 18 (1): 78–94. 153 (3), 727–750. Hendricks KB, Singhal VR. (2005). Association Between Dessouky M., Kijowski B., Verma S. (1999) Simultaneous Supply Chain Glitches and Operating Performance. batsching and scheduling for chemical processes with Management Science, 51(5):695–711. earliness and tardiness penalties, Production and Opera Hoberg K, Bradley JR, Thonemann UW. Analyzing the effect tions Management, 8(4), 433 444. of the inventory policy on order and inventory variability Disney SM, Towill DR (2002) A discrete linear control the with linear control theory. Eur J of Operational Research, ory model to determine the dynamic stability of vendor 2007;176:1620 1642. managed inventory supply chains. Int J Prod Res Holt CC, Modigliani F., Muth JF, Simon HA (1960). Plan 40(1):179–204 ning production, inventories and work force. Englewood Disney SM, Towill DR, Warburton RDH (2006) On the Cliffs, New Jersey: Prentice Hall. equivalence of control theoretic, differential, and differ Hwang, C.L., Fan, L.T., Erikson LE. (1967). Optimum pro ence equation approaches to modeling supply chains. Int duction planning by the maximum principle. Man Sci, J Prod Econ 101:194–208. 13(9), 751 55. Dolgui, A., J.M. Proth, “Supply Chains Engineering: useful Ivanov D (2010) Adaptive aligning of planning decisions on methods and techniques”, Springer, Berlin, 2010. supply chain strategy, design, tactics, and operations. Int Dolgui, A., Jean Marie Proth: Pricing strategies and models. J Prod Res 48(13):3999 4017. Annual Reviews in Control 34(1): 101 110 (2010) Ivanov D, Arkhipov A, Sokolov B (2007) Intelligent plan Fan LT, Wang CS (1964). The discrete maximum principle – ning and control of manufacturing supply chains in vir a study of multistage systems optimization. NY: Wiley. tual enterprises. Int J Manuf Tech Manag 11(2):209 227. Feichtinger G., Hartl R. (1985) Optimal pricing and produc Ivanov D, Sokolov B (2010) Adaptive Supply Chain Man tion in an inventory model. European Journal of Opera agement. Springer, London et al. tional Research, 19; 45 56. Ivanov D, Sokolov B (2011) Dynamic Supply Chain Sched Fleming WH., Rishel RW. (1975) Deterministic and stochas uling. Journal of Scheduling, DOI: 10.1007/s10951 010 tic optimal control. Springer, Berlin, Heidelberg. 0189 6. Gaimon C. (1988) Simultaneous and dynamic price, produc Ivanov, D. “DIMA – A Research Methodology for Compre tion, inventory and capacity decisions, European Journal hensive Multi Disciplinary Modelling of Production and of Operational Research, 35; 426 441. Logistics Networks”, Int J Prod Res, Vol. 47, No. 5, Goettlich, S., M. Herty, C. Kirchner, A. Klar, “Optimal con 2009, pp. 1133 1155. trol for continuous supply network models”, Networks Ivanov, D., Sokolov, B., Kaeschel, J. 2010. A multi structural and Heterogeneous Media, Vol. 1, No. 4, 2007. framework for adaptive supply chain planning and op Graves, S.C. and Willems, S.P. (2005), ‘Optimizing the sup erations with structure dynamics considerations. Euro ply chain configuration for new products’, Management pean Journal of Operational Research, 200(2); 409 420. Science, Vol. 51, No. 8, pp. 1165 1180. Jaraman, V., R. Klassen, J.D. Linton, “Supply chain man Grubbström, R.W. and Wikner, J. (1996) Inventory trigger agement in sustainable environment”, Prod Oper Manag. policies developed in terms of control theory. Interna , 25(6), pp. 1071 1074 (2007). tional Journal of Production Economics. 45: 397 406. Kannegiesser, M., Günther, H. O., van Beek, P., Grunow, M., Habla, C., Value chain management for commodi
432
Preprints of the 18th IFAC World Congress Milano (Italy) August 28 - September 2, 2011
ties: a case study from the chemical industry. OR Spec Moiseev, N.N. (1974). Element of the Optimal Systems The trum, 31 (2009), 63 93. ory. Nauka, Moskow. Khmelnitsky E., Kogan K., Maimom O. (1997). Maximum Narendra K.S. 2005. Stable adaptive systems (2 nd edition). principle based methods for production scheduling with Dover. partially sequence dependent setups. Int J Prod Res, Vol. Okhtilev, M., Sokolov, B., Yusupov, R. 2006. Intelligent 35, No. 10, 1997, pp. 2701 2712. technologies of monitoring and intellectual control. Kinemia J.G ., Gershwin S.B. (1983) An algorithm for the Moskau, Nauka (in Russian). computer control of a flexible manufacturing system. IIE Ortega, M. and Lin, L. (2004) Control theory applications to Transactions, 15, 353 362. the production inventory problem: a review. Internation Kleindorfer, P.R., Saad. G.H. (2005), “Managing Disruption al Journal of Production Research. 42: 2303 2322. Risks in Supply Chains”, Production and Operations Ouyang Y, Li X. (2010). The bullwhip effect in supply chain Management, 14(1), pp. 53 68. networks. European Journal of Operational Research, Kogan K., Herbon A. (2008) A supply chain under limited 201(3):799–810. time promotion: The effect of customer sensitivity. Eu Ouyang, Y., 2007. The effect of information sharing on sup ropean Journal of Operational Research 188 (2008) 273– ply chain stability and the bullwhip effect. Eur J of Op 292 erational Research 182 (3), 1107–1121. Kogan K., Khmelnitsky E. (2000). Scheduling: control based Pekelman, D., 1974. Simultaneous price production deci theory and polynomial time algorithms. Dordrecht, Klu sions. Operations Research 22 , pp. 788–794. wer Perea, E., Grossmann, I., Ydstie, E. and Tahmassebi, T. Kok de AG, Graves SC. Supply Chain Management: Design, (2000) Dynamic modeling and classical control theory Coordination and Operation, Amsterdam: Elsevier; 2004 for supply chain management. Computers and Chemical Kouvelis, P., Chambers, C., Wang, H. (2006), “Supply Chain Engineering. 24:1143 1149. Management Research and Production and Operations Pontryagin, L.S., Boltyanskiy, V.G., Gamkrelidze, R.V., and Management: Review, Trends, and Opportunities”, Mishchenko, E.F. (1964). The mathematical theory of Production and Operations Management, Vol. 15, Nr. 3, optimal processes. Pergamon Press, Oxford. pp. 449 469. Puigjaner L, Lainez JM (2008) Capturing dynamics in inte Krajewski, L., Wei, J.C. and Tang, L L. (2005), “Responding grated supply chain management. Comput Chem Eng to schedule changes in build torder supply chains”, 32:2582–2605. Journal of Operations Management, Vol. 23, pp. 452 69. Sarimveis H, Patrinos P, Tarantilis CD, Kiranoudis CT. Dy Kreipl, S., Pinedo, M. (2004). Planning and Scheduling in namic modeling and control of supply chain systems: A Supply Chains: An Overview of Issues in Practice. Pro review. Computers & Operations Research 2008; 35: duction and Operations Management, 13(1), 77 92. 3530 3561. Kuehnle H (2008) A system of models contribution to pro Sarker, R. And Diponegoro, A. (2009). Optimal production duction network (PN) theory. J Intell Manuf 18(5):543– plans and shipment schedules in a supply chain system 551 with multiple suppliers and multiple buyers. European Lalwani, C.S., Disney, S., Towill, D.R. (2006). Controllable, Journal of Operation Research, 194(3), 753 773. observable and stable state space representations of a Schwartz, JD, Rivera DE (2010). A process control approach generalized order up to policy. International Journal of to tactical inventory management in production Production Economics, 101, 172 184. inventory systems. International Journal of Production Lee EB, Markus L (1967) Foundations of optimal control Economics, 125(1):111 124. theory. John Wiley & Sons, New York Seirstad A., Sydsaeter K. Optimal control theory with eco Lee, H., Ö. Özer, “Unlocking the value of RFID”, Prod. nomic applications. North Holland: Amsterdam et al., Oper. Manag., 16(1), pp. 40 64, 2007. 1985 Lee, H.L. (2004), “The triple A supply chain”, Harvard Sethi, S.P., Thompson, G.L. (1981). Optimal Control Theory: Business Review, October, pp. 102 12. Applications to Management Science and Economics. Leigh JR. 2004. Control theory (2 nd Edition). London: Insti Springer, Berlin. tute of electrical engineers. Sethi, S.P., Thompson, G.L. (2006). Optimal control theory: Levin, Y., McGill, J., and Nediak, M. (2008) Risk in Reve Applications to Management Science and Econom ics, nue Management and Dynamic Pricing (pdf). Operations Second Edition. Springer, Berlin. Research, 56(2), 326 343 Shah, N., 2005. Process industry supply chains: advances and Mayne DQ, Rawlings JB, Rao CV, Scokaert POM. (2000). challenges. Computers and Chemical Engineering 29, Constrained model predictive control: Stability and op 1225–1235. timality. Automatica, 36:789 814. Sheffi, Y. (2005), The resilient enterprise, MIT Press, Massa Meepetchdee, Y. and Shah, N. (2007) ‘Logistical network chusetts. design with robustness and complexity considerations’. Siliak DD (1990) Decentralized control of complex systems. International Journal of Operations and Production Man Academic Press, New York. agement, Vol. 37, No. 3, pp. 201 222. Simchi Levi et al. (2003) Designing and managing supply Mesarovic, M.D, Takahara, Y.: General systems theory: chains. McGraw Hill. mathematical foundations. Academic Press, New York, San Francisco, London (1975)
433
Preprints of the 18th IFAC World Congress Milano (Italy) August 28 - September 2, 2011
Simchi Levi, D., Wu, S.D. and Zuo Yun, S. (2004). Hand book of quantitative supply chain analysis. Springer, New York. Skurihin V.I., Zabrodsky V.A., Kopeychenko Yu.V., 1989 Adaptive control systems in machine building industry. Moscow, Mashinostroenie, (in Russian). Son Y J, Venkateswaran J. Hierarchical supply chain plan ning architecture for integrated analysis of stability and performance. International Journal of Simulation and Process Modelling 2007;3 (3); 153 169. Stevenson, M., Spring, M. (2007), “Flexibility from a supply chain perspective: definition and review”, International Journal of Operations & Production Management ,Vol. 27, No. 7, pp. 685 713. Surana, A., Kumara, S., Greaves, M., Raghavan, U.N. Sup ply chain networks: a complex adaptive systems per spective. International Journal of Production Research 2005, Vol. 43, No. 20, pp. 4235 4265. Swaminathan JM, Smith SF and Sadeh NM (1998) Modeling supply chain dynamics: a multiagent approach. Decis Sci 29(3):607–632 Tabak D., Kuo B.C. Optimal control by mathematical pro gramming. Prentice Hall, NY, 1971. Vidal, C., Goetschalckx, M (1997) Strategic production distribution models: a critical review with emphasis on global supply chain models. Eur J Oper Res: 1 18. Wadhwa, S., Saxena, A., Chan, F.T.S. (2008), “Framework for flexibility in dynamic supply chain management”, In ternational Journal of Production Research, Vol. 46, No. 6, pp. 1373 1404. Wang, W., D. Rivera, K.G. Kempf, “Model predictive con trol strategies for supply chain management in semicon ductor industry”, Int J Prod Econ, Vol. 107, 2007, pp. 56–77. Warburton RDH, Disney SM, Towill DR, Hodgson JPE (2004) Further insights into "The stability of supply chains". Int J Prod Res 42(3):639 648 Wikner, J., Towill, D.R., Naim, M.M., 1991. Smoothing sup ply chain dynamics. International Journal Production Economics 22 (3), 231–248.
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