Hierarchical production and setup scheduling in stochasticmanufacturing systems

This paper is concerned with an asymptotic analysis of hierarchical production and setup scheduling in a stochastic manufacturing system consisting of a single failure-prone machine and facing constant demands for a number of products. At any given time the system can only produce one type of product, and the system requires a setup if production is to be switched from one type of product to another. A setup may involve setup time or setup cost or both. The objective of the problem is to minimize the total costs of setup, production, and surplus. The control variables are a sequence of setups and a production plan. An asymptotic analysis with respect to increasing rates of change in machine states gives rise to a deterministic limiting optimal control problem in which there is a control variable associated with each of the machine states and the production rate is obtained by weighting these controls with the stationary probabilities of the corresponding states. Asymptotic optimal controls for the original problem from optimal or near-optimal controls for the limiting problem are constructed     

A numerical method in optimal production and setup scheduling ofstochastic manufacturing systems

In this paper, we consider optimal production and setup scheduling in a failure-prone manufacturing system consisting of a single machine. The system can produce several types-of products, but at any given time it can only produce one type of product. A setup is required if production is to be switched from one type of product to another. The decision variables are a sequence of setups and a production plan. The objective of the problem is to minimize the cost of setup, production, and surplus. An approximate optimality condition is given together with a computational algorithm for solving the optimal control problem     

Switching curve policy for manufacturing systems with no backlogpermitted

For failure-prone manufacturing systems with deterministic machine up time, the switching curve policy, which has a special property and can be characterized by a single parameter, was shown to be optimal. In previous work, the problem of optimal production rate control was studied under the assumption that backlog is allowed. In this paper, the authors consider manufacturing systems in which backlog is not permitted. The authors show that the switching curve policy is still optimal in this case. They also consider the system model in which one machine has three states: down, idle, and operational. This model can reduce maintenance times of the machine, and then reduce its maintenance costs     

Optimal threshold control in discrete failure-prone manufacturingsystems

Bielecki and Kumar (1988) show that the threshold or hedging-point production policies are optimal in a continuous manufacturing system, even if production ability is uncertain. Their analysis assumes constant demand and processing time. In this paper, we consider a discrete manufacturing system in which production capacity, demand, and processing time are all nondeterministic. We formulate the problem into a discrete Markovian production model, and explore the most cost-effective control policy for such a system. With two more sources of uncertainty, we find that the threshold control policies are optimal among all feasible policies when the long run average cost is to be minimized. This extends Bielecki and Kumar's result which shows that the threshold policies are optimal among a subset of feasible policies     

Stohastic optimal production control problem with corrective maintenance

We consider a production control problem in a manufacturing system with failure-prone machines and a constant demand rate. The objective is to minimise a discounted inventory holding and backlog cost over an infinite planning horizon. The availability of the machines is improved through a corrective maintenance strategy. The decision variables are the production and the machine repair rates, which influence the inventory levels and the system capacity, respectively. It is shown that, for constant demand rates and exponential failure and repair times distributions of the machines, the hedging point policy is optimal. Such a policy is modified herein and parameterised by factors representing the thresholds of involved products and switching inventory levels for corrective maintenance. With the obtained policy, simulation experiments are combined to experimental design and response surface methodology to estimate the optimal production and corrective maintenance policies, respectively. The usefulness of the proposed approach is illustrated through a numerical example.     

Scheduling one-part-type serial manufacturing system under periodic demand: a solvable case

The paper studies one-part type, multiple-stage production system with periodic demands. A buffer of infinite capacity is placed after each machine. Inventory flow through buffers is controlled by machine production rates. The objective is to find a cyclic production rate, which minimizes all inventory-related expenses over an infinite planning horizon. With the aid of the maximum principle, optimal production policies are derived and the continuous-time scheduling problem is reduced to a discrete timing problem. As a result, a polynomial-time algorithm is suggested to calculate the optimal production rate. A numerical example is used to illustrate the algorithm.Scope and purposeNumerical and heuristic approaches have been suggested for production control of automated-serial-manufacturing systems. These approaches try to derive production control policies that would minimize overall costs related to inventory, backlog, and production. The quality of these approaches is often difficult to assess, and they can be time-consuming to implement. Therefore, increasing attention has been directed to optimal control policies of production systems that can be derived precisely and quickly. This paper addresses a special case of the production system manufacturing a single product type to meet a periodic demand. Given a certain assumption on cost relationship, we derive a fast and simple scheduling algorithm that calculates the optimal policy.     

Kalman filter based production control of a failure-prone single-machine single-product manufacturing system with imprecise demand and inventory information

An adaptive production control structure for failure-prone manufacturing systems under inventory and demand uncertainty is proposed. It contains estimation and forecasting modules incorporated into a control loop. The customer demand is unknown and its rate is composed of ramp-type, seasonal and random components. Information available to decision maker consists of imprecise inventory records, and the Kalman filter technique is used for estimating the inventory level and demand rate online from noisy inventory measurements. Estimates obtained are shown to converge to the actual values in stochastic sense. They are subsequently used for demand component forecasting, once the estimation errors become sufficiently small. A forecasting algorithm allows estimating ramp-type and seasonal demand components, together with their potential errors. Obtained estimates are incorporated into production control procedures, recently developed for manufacturing systems under variable and uncertain demand. Optimality conditions in the form of Hamilton-Jacobi-Bellman equations are obtained. A constructive numerical method for computing sub-optimal production policies is proposed and validated through numerical simulations.     

Optimization of the finite production rate model with scrap,rework and stochastic machine breakdown

This study employs mathematical modeling along with a recursive searching algorithm to determine the optimal run time for an imperfect finite production rate model with scrap, rework, and stochastic machine breakdown. In real-life manufacturing systems, generation of defective items and machine breakdown are inevitable. The objective of this paper is to address these issues and to be able to derive the optimal production run time. It is assumed that the proposed manufacturing system produces defective items randomly, a portion of them is considered to be scrap, and the other portion can be repaired through rework. Further, the proposed system is subject to random breakdown and when it occurs, the abort/resume (AR) policy is adopted. Under such an inventory control policy, the production of the interrupted lot will be resumed immediately when machine is fixed and restored. Mathematical modeling along with a recursive searching algorithm is used for deriving the replenishment policy for such a realistic production system.     

Adaptive control for manufacturing systems using infinitesimalperturbation analysis

Combining infinitesimal perturbation analysis (IPA) with stochastic approximation gives identification algorithms to estimate the optimal threshold value for failure-prone manufacturing systems consisting of one machine producing one part type. Two adaptive control schemes are proposed. The adaptive control schemes do not require the knowledge of the distribution functions of the up and down times. Under some appropriate conditions, the strong consistency, as well as the convergence rates, of the identification algorithms and the cost function is established for the adaptive control schemes. In particular, it is shown that central limit theorems hold for the identification algorithms     

Optimal control of pull manufacturing systems

We consider the problem of optimal control of pull manufacturing systems. We study a fluid model of a flow shop, with buffer holding costs nondecreasing along the route. The system is subject to a constant exogenous demand, thus incurring additional shortfall/inventory costs. The objective is to determine the optimal control for the production rate at each machine in the system. We exhibit a decomposition of the flow shop into “sections” of contiguous machines, where, in each section, the head machine is the bottleneck for the downstream system. We exhibit the form of an optimal control and show that it is characterized by a set of “deferral times”, one for each head machine. Machines which are upstream of a head machine simply adopt a “just-in-time” production policy. The head machines initially stay idle for a period equal to their deferral time and thereafter produce as fast as possible, until the initial shortfall is eliminated. The optimal values of these deferral times are simply obtained by solving a set of quadratic programming problems. We also exhibit special cases of re-entrant lines, for which the optimal control is similarly computable     

Optimal Production Planning in a Multi-Product Stochastic Manufacturing System with Long-Run Average Cost

This paper is concerned with the problem of production planning in a flexible manufacturing system consisting of a single or parallel failure-prone machines producing a number of different products. The objective is to choose the rates of production of the various products over time in order to meet their demands at the minimum long-run average cost of production and surplus. The analysis proceeds with a study of the corresponding problem with a discounted cost. It is shown using the vanishing discount approach for the average cost problem that the Hamilton-Jacobi-Bellman equation in terms of directional derivatives has a solution consisting of the minimal average cost and the so-called potential function. The result helps in establishing a verification theorem, and in specifying an optimal control policy in terms of the potential function. The results settle a hitherto open problem as well as generalize known results.     

Maintenance scheduling and production control of multiple-machine manufacturing systems

This paper deals with the production and preventive maintenance control problem for a multiple-machine manufacturing system. The objective of such a problem is to find the production and preventive maintenance rates for the machines so as to minimize the total cost of inventory/backlog, repair and preventive maintenance. A two-level hierarchical control model is presented, and the structure of the control policy for both identical and non-identical manufacturing systems is described using parameters, referred to here as input factors. By combining analytical formalism with simulation-based statistical tools such as experimental design and response surface methodology, an approximation of the optimal control policies and values of input factors are determined. The results obtained extend those available in existing literature to cover non-identical machine manufacturing systems. A numerical example and a sensitivity analysis are presented in order to illustrate the robustness of the proposed approach. The extension of the proposed production and preventive maintenance policies to cover large systems (multiple machines, multiple products) is discussed.     

Mathematical modeling for solving manufacturing run time problem with defective rate and random machine breakdown

This paper employs mathematical modeling for solving manufacturing run time problem with random defective rate and stochastic machine breakdown. In real life manufacturing systems, generation of nonconforming items and unexpected breakdown of production equipment are inevitable. For the purpose of addressing these practical issues, this paper studies a system that may produce defective items randomly and it is also subject to a random equipment failure. A no resumption inventory control policy is adopted when breakdown occurs. Under such a policy, the interrupted lot is aborted and malfunction machine is immediately under repair. A new lot will be started only when all on-hand inventory are depleted. Modeling and numerical analyses are used to establish the solution procedure for such a problem. As a result, the optimal manufacturing run time that minimizes the long-run average production–inventory cost is derived. A numerical example is provided to show how the solution procedure works as well as the usages of research results.     

Simulation optimization approach for hybrid flow shop scheduling problem in semiconductor back-end manufacturing

This study presents a simulation optimization approach for a hybrid flow shop scheduling problem in a real-world semiconductor back-end assembly facility. The complexity of the problem is determined based on demand and supply characteristics. Demand varies with orders characterized by different quantities, product types, and release times. Supply varies with the number of flexible manufacturing routes but is constrained in a multi-line/multi-stage production system that contains certain types and numbers of identical and unrelated parallel machines. An order is typically split into separate jobs for parallel processing and subsequently merged for completion to reduce flow time. Split jobs that apply the same qualified machine type per order are compiled for quality and traceability. The objective is to achieve the feasible minimal flow time by determining the optimal assignment of the production line and machine type at each stage for each order. A simulation optimization approach is adopted due to the complex and stochastic nature of the problem. The approach includes a simulation model for performance evaluation, an optimization strategy with application of a genetic algorithm, and an acceleration technique via an optimal computing budget allocation. Furthermore, scenario analyses of the different levels of demand, product mix, and lot sizing are performed to reveal the advantage of simulation. This study demonstrates the value of the simulation optimization approach for practical applications and provides directions for future research on the stochastic hybrid flow shop scheduling problem.     

Failure-prone production systems with uncertain demand

We consider a failure-prone manufacturing system with bursty demand arrivals. We prove that the hedging-point policy is optimal for this problem and provide analytical expressions to compute the hedging point. This allows us to compare our exact results to simpler approximations. We also show that our result leads to the solution for the constant demand rate problem, under an appropriate scaling of the demand process. We also provide a necessary and sufficient condition under which the just-in-time (JIT) policy is optimal for the case of linear, absolute value instantaneous cost     

On the ordering of optimal hedging points in a class ofmanufacturing flow control models

The optimal flow control policy of a single-product unreliable manufacturing system that must meet a constant demand rate is known to be a threshold type policy: safety production surplus levels called hedging points (thresholds) are associated with each discrete stochastic capacity state of the system and serve to protect the production process from uncertainty in future capacity availability. This correspondence extends and generalizes previous results on the ordering of optimal hedging points. The authors' method is based on examining special properties of the Bellman optimality conditions of the underlying stochastic control problem     

A time-decomposition method for sequence-dependent setup schedulingunder pressing demand conditions

This paper develops a method for continuous-time scheduling problems in flexible manufacturing systems. The objective is to find the optimal schedule subject to different production constraints: precedence constraints (bills of materials), sequence-dependent setup times, finite machine capacities, and pressing demands. Differential equations along with mixed constraints are used to model production and setup processes in a canonical form of optimal control. The proposed approach to the search for the optimal solution is based on the maximum principle analysis and time-decomposition methodology. To develop fast near-optimal solution algorithms for sizable problems, we replace the general problem with a number of sub-problems so that solving them iteratively provides tight lower and upper estimates of the optimal solution     

A simulation optimization approach in production planning of failure prone manufacturing systems

In this paper, the implementation of a new method to control the production rate of manufacturing systems, based on the combination of stochastic optimal control theory, discrete event simulation, experimental design and response surface methodology is outlined. The system under study consists of several parallel machines, multiple-product manufacturing system. Machines are subject to failures and repairs and their capacity process is assumed to be a finite state Markov chain throughout the analytical control model. The problem is to choose the production rates so as to minimize the expected discounted cost of inventory/backlog over an infinite horizon. We first show that, for constant demand rates and exponential failure and repair times distributions of the machines, the hedging point policy is optimal. The structure of the hedging point policy is then parameterized by factors representing the thresholds of involved products. With such a policy, simulation experiments are combined to experimental design and response surface methodology to estimate the optimal control policy. We obtain that the hedging point policy is also applicable to a wide variety of complex problems including non-exponential failure and repair times distributions and random demand rates. Analytical solutions may not be easily obtained for such complex situations.     

Optimal PHP production of multiple part-types on a failure-pronemachine with quadratic buffer costs

We consider a single, failure-prone machine, producing multiple part-types The objective is to minimize the expected sum of quadratic buffer costs. In general, the optimal solution to this problem is unknown. However, by restricting the allowable set of control policies to the class of prioritized hedging point (PHP) policies, we are able to determine simple, analytical expressions for the optimal hedging points. We also provide some results for choosing the optimal priority ordering     

Control of knowledgeable manufacturing cell with an unreliable agent

Aiming at the task control problems existing in the knowledgeable manufacturing system, the concept of state jump system is proposed to analyze a knowledgeable manufacturing cell with an unreliable agent. The uncertain factors of the knowledgeable manufacturing cell are addressed in the task control model by utilizing a self-learning method of probability distribution parameters of stochastic events. With the state jump system given, the task control problem is greatly simplified that the optimal task control strategy of the manufacturing cell can be obtained by the combination of the uniform technology and the stochastic dynamic programming. The objective function can be stabilized to a certain extent for different initial conditions, which verifies the feasibility of the control strategy. Compared to the random control and maximum control principles, the objective function value of the optimal control strategy in this paper is relatively low, which confirms the validity of the control strategy.