Deterministic Processing Times Research Articles

Stochastic scheduling to minimize expected maximum lateness

This paper is concerned with the problems in scheduling a set of jobs associated with random due dates on a single machine so as to minimize the expected maximum lateness in stochastic environment. This is a difficult problem and few efforts have been reported on its solution in the literature. In this paper, we first derive a deterministic equivalent to the expected maximum lateness and then propose a dynamic programming algorithm to obtain the optimal solutions. The procedures to compute optimal solutions are initially developed in the case of deterministic processing times, and then extended to stochastic processing times following arbitrary probability distributions. Moreover, several heuristic rules are suggested to compute near-optimal solutions, which are shown to be highly efficient and accurate by computer-based experiments.

Performance evaluation of SCM in JIT environment

Under stochastic demand and deterministic processing times, we discussed a single-stage JIT production system with the production-ordering and supplier kanbans and derived a probability generating function (p.g.f.) of the stationary distributions of the backlogged demand. In this paper, we extend the system to supply chain management (SCM) in JIT environment with two kinds of kanbans under stochastic demand, deterministic processing times and withdrawals with lead time. These conditions are more realistic than the previous papers. We develop an algorithm for the exact performance evaluation of the SCM such as the stationary distributions of the inventory level, production quantities and total backlogged demand in each stage, using discrete-time Markov process. Optimal numbers of two kinds of kanbans in the system are determined by minimizing a general total cost function. Numerical examples are given to show the efficiency of the proposed approach.

Scheduling jobs on a machine subject to stochastic breakdowns to minimize absolute early-tardy penalties

This article addresses the problem of scheduling n jobs with a common due date on a machine subject to stochastic breakdowns to minimize absolute early-tardy penalties. We investigate the problem under the conditions that the uptimes follow an exponential distribution, and the objective measure in detail is to minimize the expected sum of the absolute deviations of completion times from the common due date. We proceed to study in two versions (the downtime follows an exponential distribution or is a constant entailed for the repeat model job), one of which is the so-called preempt-resume version, the other of which is the preempt-repeat version. Three terms of work have been done. (i) Formulations and Preliminaries. A few of necessary definitions, relations and basic facts are established. In particular, the conclusion that the expectation of the absolute deviation of the completion time about a job with deterministic processing time t from a due date is a semi-V-shape function in t has been proved. (ii) Properties of Optimal Solutions. A few characteristics of optimal solutions are established. Most importantly, the conclusion that optimal solutions possess semi-V-shape property has been proved. (iii) Algorithm. Some computing problems on searching for optimal solutions are discussed.

Availability and throughput of unreliable, unbuffered production lines with non-homogeneous deterministic processing times

This paper studies production lines composed of several serial machines which are subject to random operation-dependent failures. The production lines have no intermediate buffers between adjacent machines. Machines have different deterministic processing times. The purpose of this paper is to propose analytical models to assess steady-state availability and throughput of such lines. Thousands of production line configurations have been experimented to compare the performance of the proposed approach to approximate existing techniques. A general simulation model was developed and statistical tests carried out to prove that the proposed approach is exact and robust to model unreliable, unbuffered, and non-homogeneous production lines.

Optimal preemptive scheduling on uniform machines with discounted flowtime objectives

We consider the problem of scheduling n jobs on m parallel machines. Each job has a deterministic processing time and a weight associated with it. For uniform machines we show that discounted flowtime is minimized by serving jobs preemptively in increasing order of their remaining processing times, assigning the job with the shortest remaining processing time to the fastest available machine.

ANALYTICAL PERFORMANCE EVALUATION OF SMALL FLOW LINES WITH SHARED BUFFER

The paper presents an approximate analytical method, based on decomposition techniques, that assesses the physical performance of small flow lines with both dedicated and shared buffer. The aim of the analytical method is to capture the interdependent behaviour of the machines in the line due to the shared buffers. The method deals with discrete and deterministic processing time, limited buffer capacity, and both time to repair (TTR) and time between failure (TBF) follow a geometric distribution. The accuracy of the analytical solutions is assessed with respect to results provided by simulation.

A new approach to two-machine flow shop problem with uncertain processing times

This paper deals with the problem of optimization of job sequence in a two-machine flow shop problem in the presence of uncertainty. It is assumed that the processing times of jobs on the machines are described by triangular fuzzy sets. A new optimization algorithm based on Johnson”s algorithm for deterministic processing times and on an improvement of McCahon and Lee”s algorithm is developed and presented. In order to compare fuzzy processing times, McCahon and Lee use mean values of their corresponding fuzzy sets. It is shown that this approach cannot fully explore possible relationships between fuzzy sets. In order to overcome this drawback we consider different fuzzy sets determined by λ-cuts of the fuzzy processing times. Extensive experiments show that the new algorithm gives better solutions with respect to makespan than existing McCahon and Lee's algorithm.

Two-moment approximations for analysing a series of automated workstations

In this research we develop two-moment approximation equations for the analysis of a series of automated workstations with infinite buffer capacity. An automated workstation is a server with deterministic processing times that experiences random operating times between failures, and then subsequent random repair times. The distribution of job inter-departure times from a workstation receiving input from an automated workstation is derived. From this distribution, expressions for the mean and variance of job inter-departure times from a workstation are obtained. These are then used as a basis for ‘linking’ workstations in a two-workstation aggregation approach. This method is compared to modelling such a system as a series of G/G/1 queues. The analysis method is easily implemented in a spreadsheet. Extensive computational experiments demonstrate the effectiveness of the overall approach.

Raw material release time control for complex make-to-order products with stochastic processing times

This paper considers a make-to-order manufacturing system producing complex products with multiple resource constraints, multiple levels of product structures, and stochastic processing times. The objective is to find the optimal raw material release times by minimising the expected sum of work-in-progress holding costs, product earliness costs and product tardiness costs. An evolution strategy-based planning tool is developed, in which the performance measure is evaluated by stochastic discrete event simulation. The tool is applied to an illustrative example that includes wide, deep and complex product structures and to a case study using data obtained from an industrial company that manufactures capital goods. It is demonstrated that the solutions produced perform substantially better than those produced by the backward scheduling based on infinite capacity, and also significantly better than those produced by the same search method based on deterministic processing times. More benefit can be achieved if the system has a higher degree of uncertainty.

Performance evaluation of production lines with finite buffer capacity producing two different products

This paper presents an approximate analytical method for the performance evaluation of a production line with finite buffer capacity, multiple failure modes and multiple part types. This paper presents a solution to a class of problems where flexible machines take different parts to process from distinct dedicated input buffers and deposit produced parts into distinct dedicated output buffers with finite capacity. This paper considers the case of two part types processed on the line, but the method can be extended to the case of n part types. Also, the solution is developed for deterministic processing times of the machines which are all identical and are assumed to be scaled to unity. The approach however is amenable of extension to the case of inhomogeneous deterministic processing times. The proposed method is based on the approximate evaluation of the performance of the k-machine line by the evaluation of 2(k-1) two-machine lines. An algorithm inspired by the DDX algorithm has been developed and the validation of the method has been carried out by means of testing and comparison with simulation.

Channel coordination in an assembly system facing uncertain demand with synchronized processing time and delivery quantity

Supply chains need to be synchronized to increase system efficiency. With increasing globalization, an assembly or a manufacturing firm may receive supplies from all over the world. Synchronization of such a system becomes difficult for two reasons: first, each supplier has certain capacity and can supply materials with a fixed but different order processing time; second, each supplier has a different cost structure and, therefore, production quantities differ due to incentives conflicts, which leads to local optimization of profits by the suppliers. In the existing literature, this problem is generally tackled by focusing either on the synchronization of order processing time or on order quantity determination. However, when system coordination is of the essence, simultaneous consideration of both becomes imperative. A model to synchronize the assembly process is proposed and the optimal order quantities in a two-echelon assembly system with one assembler (facing a stochastic demand) and multiple suppliers with deterministic order processing times is determined. A numerical study conducted shows that risk sharing and proper safety stock placement lead to better system coordination and improve system performance.

Makespan estimation and order acceptance in batch process industries when processing times are uncertain

Batch process industries are characterized by complex precedence relationships between operations, which renders the estimation of an acceptable workload very difficult. A detailed schedule based model can be used for this purpose, but for large problems this may require a prohibitive large amount of computation time. We propose a regression based model to estimate the makespan of a set of jobs. We extend earlier work based on deterministic processing times by considering Erlang-distributed processing times in our model. This regression-based model is used to support customer order acceptance. Three order acceptance policies are compared by means of simulation experiments: a scheduling policy, a workload policy and a regression policy. The results indicate that the performance of the regression policy can compete with the performance of the scheduling policy in situations with high variety in the job mix and high uncertainty in the processing times.

Performance evaluation of continuous production lines with machines having different processing times and multiple failure modes

This paper presents an approximate analytical method for the performance evaluation of asynchronous production lines with deterministic processing times, multiple failure modes and finite buffer capacity. The discrete flow of parts is approximated by a continuous flow of material. The technique developed here is based on the approximate evaluation of a K machine line by the evaluation of K−1 two-machine lines: unlike previous work on this subject, the multiple failure approach allows to consider real and virtual failure modes for each building block in which the original line is decomposed. Indeed, to reproduce in each two-machine lines the flow of material observed in the related buffer of the original line, real failures are used to take into account the reliability of the physical machines while virtual failures are introduced to model the different interruptions of flow that can arise inside the line thus preventing machines from working. To assess the accuracy of the proposed method, extensive simulation and numerical experiments have been carried out.

Parametric design and optimization for a nonlinear precision X-Y microstage

We discuss the one-machine-model with deterministic processing times. In the first part we restrict the model to the common due date case. In the second part an equivalence between this model and the ‘Equal Slack Due Date’ model is shown. As a penalty function F we use the sum of identical functions f of the lateness (both earliness and tardiness). The function f must be monotonous with respect to the absolute lateness. No other restrictions are necessary. Several special cases are known in the literature and summarized here to show the generality of our approach. This idea of penalizing jobs whose completion times deviate from their due date arises in some scheduling models for ‘just-in-time’ production. We give two structural results for an optimal schedule: An optimal schedule works without idle times except eventually one at time 0 and is ‘v-shaped’. Using these two structural results we construct an optimal schedule with a pseudopolynomial algorithm. This algorithm has a complexity bound of O(nP), where P denotes the sum of the given processing times. For a subclass of penalty functions, we extend the pseudopolynomial algorithm to a fully polynomial approximation scheme. The complexity bound for this approximation scheme is of the order O(n3·(1/ε)), where ε denotes the relative precision of the approximate solution. As mentioned above we show the equivalence of the ‘common due date’ problem and the ‘equal slack due date’ problem. Mathematically these two problems can be expressed as:Common Due Date Rule: d(i) := d for all jobs i, and Equal Slack Rule: d(i) := d + p(i) for all jobs i, where p(i) denotess the given processing time of job i and d is a constant value.

AN INTEGRATED ANALYTICAL/SIMULATION APPROACH FOR ECONOMIC DESIGN OF AN AGV SYSTEM

This paper proposes an integrated analyticd/simulation approach for designing an automated guided vehicle system (AGVS) which consists of AGVs, machines with input buffers and a dispatching station in a just-in-time (JIT) envkonment. The objective is to determine the number of AGVs, the input buffer capacities and locations of the machines that minimize a cost function under the constraint that the planned utilization of each machine is achieved. The integrated analytical/simulation approach employs a simulation model to evaluate the performance of the AGVS and an analytical approach to reduce the repetition number of simulations in searching an optimal solution. The analytical approach leads to an efficient iterative procedure based on monotonicity properties of the cost function and the machine utilization in each design factor, and lower bounds of the number of AGVs and the input buffer capacities. Moreoverl initial locations of the machines are derived from the HLP inequality. Computational results are given to demonstrate the efficiency of the proposed procedure. It is observed that the lower bounds md the initial locations aze the optimal solution in case of deterministic processing times at machines. Traditional approaches to modeling manufacturing systems can be categorized as either ana- lytical based or simulation based (Mahadevan and Narendran 1993). Analytical approaches which model the systems by means of mathematical equations often need unrealistic assump- tions, while simulation models are often time-consuming and do not provide exact solutions. Shantikumar and Sargent (1983) defined four hybrid analyticd/simulation models. Their analytical and simulation models are developed independently, and their corresponding so- lution procedures are combined in problem solving. Class four type in their hybrid models is defined as a model in which a simulation model is used as an overall model of the total system, and it requires values from the solution procedure of the analytic mode1 of a por- tion of the system for some or all of its parameters. Most of conventional hybrid approaches employ a simulation model as an overall model of the total system and an analytical model only for determining some of its initial parameters of the simulation model. However, the conventional approaches are not sufficient to reduce the repetition number of simulations after setting the initial parameters- In many problems of designing manufacturing systems7 the cost functions and constraints have monotone structure in design parameters. For example7 as the number of facilities increases, both the total cost and production rate of the system increase. As the buffer size between machines increases, the production rate of the system increases. As the performance of facilities becomes higher, the rnakespan becomes shorter. Such monotone structure has been considered in analytical approaches (Buzacott and Shanthikumar 1993? Glasserrrian and Yao 1994)- This paper proposes an integrated analytical/simulation approach which

OPTIMAL BUFFER ALLOCATION IN ASYNCHRONOUS CYCLIC MIXED‐MODEL ASSEMBLY LINES

This paper considers a problem of optimal buffer allocation in cyclic asynchronous mixedmodel assembly lines with deterministic processing times. An analytical model is used to provide new insights into properties of optimal buffer allocation, that is, a buffer configuration that guarantees the highest possible throughput rate on the assembly line with a minimum number of buffers. Optimal buffer configuration is characterized, and an efficient algorithm to find such a configuration is developed. The approach proposed in this paper also provides insights on how to allocate a given number of buffers to workstations on the assembly line to maximize the throughput rate.

An analytical formula for variance of output from a series-parallel production system with no interstation buffers and time-dependent failures

This paper presents a method to determine the mean and the variance of the amount of materials produced in a fixed time interval by a continuous materials flow production system with N stations in series and M stations in parallel and no interstation buffers. Unreliable stations with exponential failure and repair times, time dependent failures, and deterministic processing times are considered. Closed-form expressions for the asymptotic mean and variance of the amount of materials produced per unit time are given for series, parallel, and series-parallel production systems with identical stations. It is shown that the distribution of the amount of materials produced in a fixed time is asymptotically normal. By using this property, effects of variability on the due-date performance are investigated by considering the probability of meeting a customer order on time. Numerical experiments that explore some relationships among performance measures and production system parameters are also presented.

A Method for Performance Evaluation of Automated Flow Lines

In order to find the optimal configuration of an automated flow line it is necessary to compare the performance of several alternative configurations. In this situation the use of simulation is normally not viable given the high modelling and computational effort it would require; on the other hand, the application of classical analytical tools would often entail the adoption of unrealistic assumptions. To solve this problem, the paper presents an approximate analytical method that overcomes many of the limitations of analytical tools while preserving good accuracy and low computational effort. In particular the method can deal with deterministic processing times, finite buffer capacity and multiple failure modes of the machines. This last feature represents an important improvement over existing techniques and is very useful when dealing with automated assembly lines where each station is composed by different devices that have different reliability and different ways to get repaired. The validation of the method has been carried out by means of extensive testing and comparison with simulation. An application to a real case is also reported.

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.

A generalized Johnson's rule for stochastic assembly systems

Johnson's rule gives the optimal order to schedule a set of jobs through a two-machine flow shop with deterministic processing times. We extend Johnson's rule to fork-join systems with random processing times. Most of the system may be an arbitrary network. We give conditions under which Johnson's rule is optimal in the stochastic sense, in the increasing convex sense, and in expected value. © 1997 John Wiley & Sons, Inc. Naval Research Logistics 44: 211–220, 1997