Hierarchical Decision Making in Stochastic Manufacturing Systems

One of the most important methods in dealing with the optimization of large, complex systems is that of hierarchical decomposition. The idea is to reduce the overall complex problem into manageable approximate problems or subproblems, to solve these problems, and to construct a solution of the original problem from the solutions of these simpler prob­ lems. Development of such approaches for large complex systems has been identified as a particularly fruitful area by the Committee on the Next Decade in Operations Research (1988) [42] as well as by the Panel on Future Directions in Control Theory (1988) [65]. Most manufacturing firms are complex systems characterized by sev­ eral decision subsystems, such as finance, personnel, marketing, and op­ erations. They may have several plants and warehouses and a wide variety of machines and equipment devoted to producing a large number of different products. Moreover, they are subject to deterministic as well as stochastic discrete events, such as purchasing new equipment, hiring and layoff of personnel, and machine setups, failures, and repairs.

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I. Introduction and Models of Manufacturing Systems.- 1. Concepts of hierarchical decision making.- 1.1 Introduction.- 1.2 A brief review of the literature.- 1.3 Our approach to hierarchical decision making.- 1.4 Plan of the book.- 2. Models of manufacturing systems.- 2.1 Introduction.- 2.2 A parallel-machine, single product model.- 2.3 A parallel-machine, multi-product model.- 2.4 A single product dynamic flowshop.- 2.5 A dynamic jobshop.- 2.6 Setup costs and setup times.- 2.7 Interaction with other functional areas.- II. Optimal Control of Manufacturing Systems: Existence and Characterization.- 3. Optimal control of parallel machine systems.- 3.1 Introduction.- 3.2 Problem formulation.- 3.3 Properties of the value function.- 3.4 Turnpike sets with constant demand.- 3.5 Turnpike sets with reliable machines.- 3.6 Miscellaneous results.- 3.7 Notes.- 4. Optimal control of dynamic flowshops.- 4.1 Introduction.- 4.2 Problem formulation.- 4.3 Properties of the value function.- 4.4 An equivalent deterministic problem.- 4.5 HJBDD equations and boundary conditions.- 4.6 Extension of results to dynamic jobshops.- 4.7 Notes.- III: Asymptotic Optimal Controls.- 5. Hierarchical controls in systems with parallel machines.- 5.1 Introduction.- 5.2 Formulation of the single product case.- 5.3 Properties of the value function.- 5.4 The limiting control problem.- 5.5 Convergence of the value function.- 5.6 Asymptotic optimal feedback controls.- 5.7 A simple example.- 5.8 Capacity dependent on production rate.- 5.9 Machine states with weak and strong interactions.- 5.10 The multi-product case.- 5.11 Concluding remarks.- 5.12 Notes.- 6. Hierarchical controls in dynamic flowshops.- 6.1 Introduction.- 6.2 Formulation of the m-machine flowshop.- 6.3 A special case: The two-machine flowshop.- 6.4 Construction of candidate open-loop controls.- 6.5 Preliminary results.- 6.6 Proof of asymptotic optimality.- 6.7 Concluding remarks.- 6.8 Notes.- 7. Hierarchical controls in dynamic jobshops.- 7.1 Introduction.- 7.2 A graph-theoretic framework.- 7.3 The optimization problem.- 7.4 Lipschitz continuity.- 7.5 Asymptotic optimal open-loop controls.- 7.6 Concluding remarks.- 7.7 Notes.- 8. Hierarchical production and setup scheduling in a single machine system.- 8.1 Introduction.- 8.2 Problem formulation.- 8.3 HJB equations.- 8.4 The limiting decision problem.- 8.5 Asymptotic optimal open-loop decisions.- 8.6 Asymptotic optimal feedback decisions.- 8.7 A simple example.- 8.8 Concluding remarks.- 8.9 Notes.- 9. Hierarchical feedback controls in two-machine flowshops.- 9.1 Introduction.- 9.2 Original and limiting problems.- 9.3 An explicit solution of the limiting problem.- 9.4 An asymptotic optimal feedback control.- 9.5 Relationship to the Kanban control.- 9.6 Concluding remarks.- 9.7 Notes.- IV: Multilevel Hierarchical Decisions.- 10. A production and capacity expansion model.- 10.1 Introduction.- 10.2 Problem formulation.- 10.3 Viscosity solutions of HJB equations.- 10.4 The limiting control problem.- 10.5 Verification theorems.- 10.6 Asymptotic optimal feedback decisions.- 10.7 Concluding remarks.- 10.8 Notes.- 11. Production-marketing systems.- 11.1 Introduction.- 11.2 Problem formulation and possible hierarchies.- 11.3 Analysis of the value functions.- 11.4 Asymptotic optimal feedback controls.- 11.5 Concluding remarks.- 11.6 Notes.- V: Computations and Conclusions.- 12. Computations and evaluation of hierarchical controls.- 12.1 Introduction.- 12.2 Problems and policies under consideration.- 12.3 Computational issues.- 12.4 Comparison of HC with other policies.- 12.5 Concluding remarks.- 12.6 Notes.- 13. Further extensions and open research problems.- 13.1 Introduction.- 13.2 Asymptotic optimal feedback controls.- 13.3 Error bounds for feedback controls.- 13.4 Average costs and robust controls.- 13.5 General systems.- 13.6 Final thoughts.- VI: Appendices.- A. Finite state Markov chains.- B. Martingale problems, tightness, and Skorohod representation.- C. Rate of convergence of Markov chains.- D. Control-dependent Markov chains.- E. Convergence of Markov chains with two parameters.- F. Convex functions.- G. Viscosity solutions of HJB equations.- H. Value functions and optimal controls.- I. A review of relevant graph theory.- J. Miscellany.- Author index.- Copyright permissions.