A multiperiod stochastic production planning and sourcing problem with service level constraints

We study a stochastic multiperiod production planning and sourcing problem of a manufacturer with a number of plants and/or subcontractors. Each source, i.e. each plant and subcontractor, has a different production cost, capacity, and lead time. The manufacturer has to meet the demand for different products according to the service level requirements set by its customers. The demand for each product in each period is random. We present a methodology that a manufacturer can utilize to make its production and sourcing decisions, i.e., to decide how much to produce, when to produce, where to produce, how much inventory to carry, etc. This methodology is based on a mathematical programming approach. The randomness in demand and related probabilistic service level constraints are integrated in a deterministic mathematical program by adding a number of additional linear constraints. Using a rolling horizon approach that solves the deterministic equivalent problem based on the available data at each time period yields an approximate solution to the original dynamic problem. We show that this approach yields the same result as the base stock policy for a single plant with stationary demand. For a system with dual sources, we show that the results obtained from solving the deterministic equivalent model on a rolling horizon gives similar results to a threshold subcontracting policy. Correspondence to: Fikri KaraesmenThe authors are grateful to Yves Dallery for his ideas, comments and suggestions on the earlier versions of this paper.     

The Simultaneous Determination of Plant and Leased Warehouse Capacities for a Firm Facing Uncertain Demand in Several Regions

For each of several regions a firm has forecasted a probability distribution of future annual demand, and the plan is to build a single production facility to serve this demand. Each region is served by a simple leased warehouse with a lease cost which is a linear function of capacity. Any regional demand exceeding warehouse capacity is lost. The cost of building the plant is a nonlinear function of its capacity, and the unit cost of producing the product and supplying it to a region is constant. We develop an efficient algorithm based on the Kuhn-Tucker conditions for simultaneously determining the plant and warehouse capacities which will maximize the firm's expected profit. The paper includes a numerical example demonstrating the use of this algorithm.     

Optimal Production Strategies for Identical Production Lines with Minimum Operable Production Rates

When demand is less than plant capacity, production must be reduced or inventories will build. One method for matching production with demand is to throttle production at a rate equal to demand. Another production control method is to run production lines at capacity and periodically shut down one or more lines. This paper develops optimal production strategies for plants with identical production lines, each of which has a minimum operable production rate.     

THE PRODUCTION PLANNING OF MANY-PRODUCT ASSEMBLY OPERATIONS

SUMMARY The paper discusses planning problems which arise when different products are manufactured on several assembly lines and production has to be co-ordinated with other manufacturing centres. In a limited labour market flexible production capacity requires the transfer and retraining of existing workers.

The first section of the paper explores the simplest possible planning element which still contains enough of the essence of the problems to shed some light on the nature of a solution, This model consists of the planning of production of a single product in one location, under the supposition that teams of girls may be relinquished to (or obtained from) a labour reservoir at a penalty which corresponds to retraining costs. A production planning game based on this model is constructed, and played with teams of human planners. The game is also played by means of two types of systematic decision rules:

(.1) a linear production smoothing rule:

(.2) the optimum rule for the game, developed by dynamic programming. This permits a comparison of the efficacy of human planners in comparison to automatic decision rules.

The dynamic programming model is extended in the next part of the paper to cover a more realistic model of the situation. The results of the dynamic programming analysis are shown to be a very valuable tool for the production planning of several product types in one plant, as well as for the co-ordination of different plants.     

Decomposition methods for structural reliability analysis revisited

This paper presents new theoretical results to demonstrate that the referential dimensional decomposition (RDD) and cut-high-dimensional model representation (cut-HDMR), each developed independently and from a distinct perspective, lead to identical function approximations. Therefore, the reliability method stemming from the cut-HDMR approximation is precisely the same as the reliability method rooted in the RDD approximation. However, a second-moment error analysis finds neither the RDD approximation nor the cut-HDMR approximation to be optimal, whereas the approximation derived from the analysis-of-variance dimensional decomposition (ADD) results in minimum error for an arbitrary truncation. The expected errors from the RDD approximations are at least four to eight times larger than the errors from the ADD approximations. Therefore, further enhancements of decomposition-based reliability methods are possible by switching from RDD to ADD approximations. For both approximations, the decomposition can be truncated by an effective superposition dimension linked to respective approximation errors.     

Lagrangean relaxation and hybrid simulated annealing tabu search procedure for a two-echelon capacitated facility location problem with plant size selection

In this paper, we study a two-echelon capacitated facility location problem with plant size selection (TECFLP-PSS). Given a set of potential sites for plants, each of which is associated with several possible sizes and corresponding unit production costs, a set of potential sites for capacitated depots and a set of customers with demands, the TECFLP-PSS aims to optimise the plant locations and sizes, the depot locations and the product flows from the opened plants to the opened depots and then to the end customers under single-source constraints. The objective is to satisfy all customers’ demands with a minimum total cost of facility opening, production and transportation. We develop a mixed integer programming model and propose a Lagrangean relaxation approach combined with new valid inequalities and core problem to achieve tight lower and upper bounds for this problem. We then improve the upper bound with a hybrid simulated annealing tabu search procedure. Computational experiments on benchmarks and randomly generated instances are conducted to validate the effectiveness and efficiency of the proposed method.     

Modelling ramp-up curves to reflect learning: improving capacity planning in secondary pharmaceutical production

The experience gained during production ramp-up leads to an increase of the effective production capacity over time. However, full utilisation of production capacity is not always possible during ramp-up. In such cases, the experience gained and hence the available effective capacity are overestimated. We develop a new method, which captures ramp-up as a function of the cumulative production volume to better reflect the experience gained while producing the new product. The use of the more accurate and computationally effective approach is demonstrated for the case of secondary pharmaceutical production. Due to its regulatory framework, this industry cannot fully exploit available capacities during ramp-up. We develop a capacity planning model for a new pharmaceutical drug, which determines the number and location of new production lines and the build-up of inventory such that product availability at market launch is ensured. Our MILP model is applied to a real industry case study using three empirically observed ramp-up curves to demonstrate its value as decision support tool. We demonstrate the superiority of our volume-dependent method over the traditional time-dependent ramp-up functions and derive managerial insights into the selection of ramp-up function and the value of shortening ramp-ups.     

Performance of infra-plant overtime policies

Production quotas smooth production in order to synchronize inter-plant production networks. To meet quotas, plants work overtime whenever they're behind. In addition to this 'mandatory' overtime required to satisfy the quota of finished product at the end of the line, many plants run selected stations within the plant overtime to fill critical buffers within the plant. We call this intra-plant overtime 'preventive' overtime because by spending a little overtime this shift on critical stations, the plant can avoid massive overtime next shift. We describe this real production control problem currently facing industry. The problem arises because of (often underestimated or ignored) production variability. We exhibit actual production data that shows the degree of variability often inherent in current manufacturing processes. We describe four policies and compare them using simulation.     

Optimal Production Strategies for Plants with Multiple Shutdown Levels and Multiple Production Lines

When demand is less than plant capacity, aggregate production and demand can be matched by (1) throttling production at a rate equal to demand or (2) producing at capacity production rates and shutting down periodically to reduce inventories. In some plants several levels of shutdown are feasible. For example, a first level of shutdown may involve only equipment, while a second level of shutdown may involve shutting down equipment and laying off workers. Moreover, when a plant has multiple production lines, each line can be throttled or periodically shut down. This paper uses classical optimization techniques to develop optimal production strategies for plants with multiple shutdown levels and multiple, identical production lines.     

Optimal design of stochastic production lines: a dynamic programming approach

We consider the problem of choosing the number and type of machines for each station in a new production line where the sequence of processes (i.e., manufacturing recipe) has already been established. We formulate a model to minimize cost (investment plus operating) subject to constraints on throughput and cycle time. Using queueing network approximations within a dynamic programming framework, we develop a line design algorithm that works in station-wise fashion. For computational tractability, we must discretize a continuous state space. However, we are able to compute bounds on the error in the cost function as a guide to the appropriate choice of grid size. We conclude by applying our algorithm to an industrial problem that motivated this work.     

Dynamic modeling of the tradeoff between productivity and safety in critical engineering systems

Short-term tradeoffs between productivity and safety often exist in the operation of critical facilities such as nuclear power plants, offshore oil platforms, or simply individual cars. For example, interruption of operations for maintenance on demand can decrease short-term productivity but may be needed to ensure safety. Operations are interrupted for several reasons: scheduled maintenance, maintenance on demand, response to warnings, subsystem failure, or a catastrophic accident. The choice of operational procedures (e.g. timing and extent of scheduled maintenance) generally affects the probabilities of both production interruptions and catastrophic failures. In this paper, we present and illustrate a dynamic probabilistic model designed to describe the long-term evolution of such a system through the different phases of operation, shutdown, and possibly accident. The model's parameters represent explicitly the effects of different components' performance on the system's safety and reliability through an engineering probabilistic risk assessment (PRA). In addition to PRA, a Markov model is used to track the evolution of the system and its components through different performance phases. The model parameters are then linked to different operations strategies, to allow computation of the effects of each management strategy on the system's long-term productivity and safety. Decision analysis is then used to support the management of the short-term trade-offs between productivity and safety in order to maximize long-term performance. The value function is that of plant managers, within the constraints set by local utility commissions and national (e.g. energy) agencies. This model is illustrated by the case of outages (planned and unplanned) in nuclear power plants to show how it can be used to guide policy decisions regarding outage frequency and plant lifetime, and more specifically, the choice of a reactor tripping policy as a function of the state of the emergency core cooling subsystem.     

Modelling and analysis of energy footprint of manufacturing systems

Increasing the energy efficiency of manufacturing plants will reduce the production costs and environmental impact. In order to analyse and improve the energy efficiency of manufacturing plants, however, we need models to evaluate the energy footprints of the plants. A key challenge of estimating plant-level footprints is that systemic methods of connecting information on the product, machine and plant levels are not available. Thus, we propose methods to parameterise product-level elements and to model machine-level factors based on those elements. From the machine-level models, the proposed approach performs simulation experiments and provides the energy footprints in closed-form equations for the plant level. We also suggest that the resulting model can be combined with probabilistic techniques to benchmark the energy efficiency of plants at the industry level. In a case study, we demonstrate how to apply the proposed methods to estimate the energy footprint of a hypothetical plant. The procedures introduced here enable manufacturers to evaluate the energy consumption of their facilities at early stages of manufacturing, and provide tools to assess the energy efficiency of their plant by comparison with peers.     

Dynamic Programming for QFD Optimization

Quality function deployment (QFD) is a useful method in product design and development and its aim is to improve the quality and to better meet customers' needs. Due to cost and other resource constraints, trade‐offs are always needed. Many optimization methods have been introduced into the QFD process to maximize customer satisfaction under certain constraints. However, current optimization methods sometimes cannot give practical optimal results and the data needed are hard or costly to get. To overcome these problems, this paper proposes a dynamic programming approach for the optimization problem. We first use an extended House of Quality to gather more information. Next, limited resources are allocated to the technical attributes using dynamic programming. The value of each technical attribute can be determined according to the resources allocated to them. Compared with other optimization methods, the dynamic programming method requires less information and the optimal results are more relevant. Copyright © 2005 John Wiley & Sons, Ltd.     

Distribution function of the shortest path in networks of queues

In this paper, we consider acyclic networks of queues as a model to support the design of a dynamic production system. Each service station in the network represents a manufacturing or assembly operation. Only one type of product is produced by the system, but there exist several distinct production processes for manufacturing this product, each one corresponding with a directed path in the network of queues. In each network node, the number of servers in the corresponding service station is either one or infinity. The service time in each station is either exponentially distributed or belongs to a special class of Coxian distribution. Only in the source node, the service system may be modeled by an queue. The transport times between every pair of service stations are independent random variables with exponential distributions. In method proposed in this paper, the network of queues is transformed into an equivalent stochastic network. Next, we develop a method for approximating the distribution function of the length of the shortest path of the transformed stochastic network, from the source to the sink node. Hence, the method leads to determining the distribution function of the time required to complete a product in this system (called the manufacturing lead time). This is done through solving a system of linear differential equations with non-constant coefficients, which is obtained from a related continuous-time Markov process. The results are verified by simulation.This complete issue was revised and published online in November 2004. The previous version contained a false date. Correspondence to: Mohammad Modarres     

Production planning in automotive powertrain plants: a case study

Based on a real case study from the automotive industry, this paper deals with production planning in powertrain plants. We present an overview of the production planning process and propose a mixed integer linear programme to determine the production quantities of each product over a planning horizon of several days. Then, using real data of an engine assembly line, we simulate the performance obtained through the proposed model within a rolling horizon planning process. We perform multiple tests in order to evaluate the impact of two parameters involved in this process: planning frequency and frozen horizon length. Furthermore, in order to illustrate the value of improving coordination between engine plants and their customers, we evaluate the impact of the quality of demand information (orders and forecasts). We analyse the simulation results and provide insights and recommendations in order to achieve a good trade-off between service level, inventory, and planning stability.     

Linking networks and plant roles: the impact of changing a plant role

Many manufacturing firms are expanding their global footprint to explore new opportunities for efficient and effective production. The strategic perspective on international manufacturing networks involves both the network level and the plant level. A key aspect is the relationship between the network and the role of plants. In this research, we investigate the relationship between the network and plant perspectives in international manufacturing networks. We use an embedded case study that includes five plants in two product networks over a period of three years. We analyse how changing the role of one plant affects the network as well as the roles of the other plants in the networks. We find that decisions on plant roles are, to a very high degree, network decisions and not decisions for individual plants. Based on the insights into the case study, we also develop a framework for mapping manufacturing networks, including market coverage, plant location and site competence.     

Estimating the distribution and variance of time to produce a fixed lot size given deterministic processing times and random downtimes

Managing production throughput variability is necessary to reduce system costs and improve throughput. Critical in this management is predicting how system changes, like reduced repair times, will change throughput variability. Many manufacturing plants produce products in fixed lot sizes to meet production targets which causes system variability to be manifested as variability in the time to produce the lot. We derive analytical approximations of the density function and variance of the time to produce a fixed lotsize on a single workstation with deterministic processing times and random downtimes. These expressions can be applied to larger production systems, by modelling the system as a single workstation with parameters that approximate the system's output. This result quantifies how workstation failures, repair times, and speed impact the density and variance of time to produce a fixed lotsize. The density function also generates a cycle time distribution when the lost size is one. This is useful in discrete event simulation.     

Decentralized multi-product multi-stage systems with backorders

A multi-stage production/inventory system with decentralized two-card kanban control policies producing multiple product types is considered. The system involves one-at-a-time processing at each stage, finite target levels in the buffers and batch transfers between production stages. The demand process for each product is assumed to be Poisson and excess demand is backordered. Products have a priority structure and the processor at each stage is shared according to a switching rule. Our objective is to obtain steady-state performance measures such as the average inventory and service levels for each product type. We propose an approximation algorithm based on: (i) characterization of the delay by a product type before receiving the processor's attention at each stage; and (ii) creation of subsystems for all the storage activity and phase-type modeling of the remaining system's behavior. Numerical examples are presented to show the accuracy of the method and its potential use in system design.     

The Choice of Manufacturing Technology in the Presence of Dynamic Demand and Experience Effects

This study investigates the intertemporal manufacturing policy of a firm faced with dynamic demand conditions and experience effects in production. We consider a demand structure characterized by a Mansfield-like product diffusion process and examine two manufacturing processes: a low-volume, high-variable, low-fixed cost function, and a high-volume, low-variable, high-fixed cost function. An investment and production model that incorporates these tradeoffs is formulated and analyzed as a two-phase optimal control problem. The major findings relate to the production rate, the duration of each manufacturing phase, and the switching policies from one phase to the other. Illustrations of the methodology are provided and. discussed.     

An original approximate method for estimating the invariant probability distribution of a large class of multi-dimensional nonlinear stochastic oscillators

This paper proposes an original method for obtaining analytical approximations of the invariant probability density function of multi-dimensional Hamiltonian dissipative dynamic systems under Gaussian white noise excitations, with linear non-conservative parts and nonlinear conservative parts. The method is based on an exact result and a heuristic argument. Its pertinence is attested by numerical tests.