Studying unbalanced workload and buffer allocation of production systems using multi-objective optimisation

Numerous studies have investigated the effects of unbalanced service times and inter-station buffer sizes on the efficiency of discrete part, unpaced production lines. There are two main disadvantages of many of these studies: (1) only some predetermined degree of imbalance and patterns of imbalance have been evaluated against the perfectly balanced configuration, making it hard to form a general conclusion on these factors; (2) only a single objective has been set as the target, which neglects the fact that different patterns of imbalance may outperform with respect to different performance measures. Therefore, the aim of this study is to introduce a new approach to investigate the performance of unpaced production lines by using multiple-objective optimisation. It has been found by equipping multi-objective optimisation with an efficient, equality constraints handling technique, both the optimal pattern and degree of imbalance, as well as the optimal relationship among these factors and the performance measures of a production system can be sought and analysed with some single optimisation runs. The results have illustrated that some very interesting relationships among the key performance measures studied, including system throughput, work-in-process and average buffer level, could only be observed within a truly multi-objective optimisation context. While these results may not be generalised to apply to any production lines, the genericity of the proposed simulation-based approach is believed to be applicable to study any real-world, complex production lines.     

Optimal composition of number and size of machines in a multi-stage make-to-order system with due dates

We develop an optimisation model to minimise costs for work-in-process (WIP), finished goods inventory (FGI), backorders, and capacity for a multi-stage production system applying a work-ahead window work release policy. Customers arrive to a make-to-order production system and have stochastic due dates. The parameters to optimise are the capacity invested at each stage, consisting of the number of machines and the processing rate (defined by a set of possible processing rates), as well as the work-ahead window. An optimality condition is provided showing when it is optimal to invest into a single machine at each stage if processing rates are continuous decision variables. The optimality of increasing capacity towards the customer end of the production line is proven under certain conditions. For a special case consisting of two stages with M/M/s queues with exponentially distributed customer required lead time, explicit expressions for WIP, FGI and backorders are developed. A set of numerical examples illustrates the influence of predefined processing rates on the optimal number and selection of machines. A simple solution heuristic for the problem with a predefined set of processing rates is proposed and its performance is shown in a numerical example. Additionally, the influence of uncertain input rates into the production system is discussed.     

A decision support framework for global supply chain modelling: an assessment of the impact of demand,supply and lead-time uncertainties on performance

We developed a decision support framework for a global manufacturer of specialty chemicals to study the relative impact of demand, supply and lead-time uncertainties on cost and customer service performance. Our approach combines optimisation and simulation methodologies as follows: mathematical models provide optimal plans via a novel approach to the supply chain planning mechanism of the Company. Simulation models execute the supply chain plans so as to allow the examination of the outcomes under the various sources of uncertainty. The iterative use of optimisation and simulation methodologies allows the user the benefit of obtaining optimal solutions while revealing the impact of uncertainties on system performance. Our results indicate that demand uncertainty has the greatest negative impact on performance for the supply chain that we modelled in this study, emphasising the importance of effective forecasting. The relative importance of supply and lead-time uncertainties varies according to the performance measures. While our results are valid for the specific supply chain and the operating environment we modelled, our study emphasises the importance of the ability to model supply chains realistically to obtain valid and useful results.     

Quality issue in forecasting problem of production and maintenance policy for production unit

In this study, we consider an unreliable deteriorating production system that produces conforming and non-conforming products to satisfy a random demand under a given service level and during a finite horizon. The production system is subjected to a failure-prone machine. The quality of the produced products is affected by the machine deterioration since the rate of defectives increases as the deterioration increases. Preventive maintenance actions can be piloted on the production system to reduce the influence of deterioration and the defective rate. A joint control policy is based on a stochastic production and maintenance planning problem with goals to determine, firstly, the economic plan of production and secondly, the optimal maintenance strategy. The proposed jointly optimisation minimises the total cost of production, inventory, maintenance and defectives. A failure rate and quality relationship are defined to show the influence of the production rates variation on the failures rate as well as on the defective rate. A numerical example and an industrial case study are adopted to illustrate the proposed approach and a sensitivity analysis to validate the jointly optimisation.     

Optimising installation (R,Q) policies in distribution networks with stochastic lead times: a comparative analysis of guaranteed- and stochastic service models

This paper studies two modelling approaches to the multi-echelon inventory optimisation problem in a distribution network with stochastic demands and lead times. It compares the performance of a novel guaranteed-service model (GSM), using an installation (R, Q) inventory control policy, with a stochastic service model (SSM) considering ordering, holding and flexibility costs. From both cycle service level and fill rate perspectives, our numerical analysis of the 1-warehouse 2-retailer network shows that cost difference between both models is driven by the internal service level at the warehouse. The GSM outperforms the SSM for over 80% of the simulated instances and realises an average total cost improvement of approximately 10%. This analysis goes against earlier results that showed a relatively low-cost difference between the two approaches, and demonstrates that it is worthwhile to evaluate competing models for multi-echelon inventory optimisation in real-world supply chains with batch ordering and variable lead times.     

Dynamic pricing,leadtime quotation and due date based priority dispatching

This study considers a profit-maximising make-to-order manufacturing firm that (i) dynamically quotes a price/leadtime pair to arriving prospective customers who then decide whether or not to place an order by trading off the price and leadtime and (ii) dispatches placed orders. We model the marketing–operations collaboration problem as a Markov decision process to obtain the optimal quotation and dispatching policy numerically. We further investigate the sub-optimality of several sequential approaches. Our numerical results show that sub-optimality is negligible when the tardiness penalty is proportional to tardiness and the customer sensitivities to price and leadtime quotes are similar. However, it is considerable when tardiness of orders is penalised with a fixed cost and the customers differ significantly in their sensitivity to price and leadtime. By joint optimisation, it is possible to make more appealing price/leadtime quotes to customers and at the same time reach a better service level. On the other hand, the joint optimisation can also suggest the lowering of a firm’s service level in order to achieve higher profits.     

Selecting machines and buffers in unreliable series-parallel production lines

This article formulates a new optimal design problem of a series-parallel manufacturing production line, where parallel machines and in-process buffers are included to achieve a greater production rate. The objective is to maximise production rate subject to a total cost constraint. Machines and buffers are chosen from a list of products available in the market. The buffers are characterised by their cost and size. The machines are characterised by their cost, failure rate, repair rate and processing time. To estimate series-parallel production line performance, an analytical decomposition-type approximation is proposed. Simulation results show that this approximate technique is very accurate. The optimal design problem is formulated as a combinatorial optimisation one where the decision variables are buffers and types of machines, as well as the number of parallel machines. To solve this problem, ant colony optimisation and simulated annealing are compared empirically through several test problems.     

Planning of capacity,production and inventory decisions in a generic reverse supply chain under uncertain demand and returns

There is a growing interest for the design and operation of reverse supply chain systems due to the cost and the legislation issues. In this paper, we address the disassembly, refurbishing and production operations in a reverse supply chain setting for modular products such as computers and mobile phones considering the uncertainties in this system, which are the return amounts of the used products and demand for final products. We develop a large-scale mixed integer programming model in order to capture all characteristics of this system, and use two-stage stochastic optimisation and robust optimisation approaches to analyse the system behaviour. In the first stage, we focus on the strategic decisions about the capacities at disassembly and refurbishing sites considering different scenarios regarding the uncertainties in the system. In the second stage, we analyse the operational decisions such as production, inventory and disposal rates. We observe through our extensive numerical analysis that the randomness of demand and return values effect the performance of the system substantially and the uncertainty of the return amounts of used products is much more important than the uncertainty of demand in this system.     

A single-vendor single-buyer production-inventory model with controllable lead time and service level constraint for decaying items

This paper presents a production-inventory model in a single-vendor single-buyer system for decaying items which deplete with constant decay rate. It is assumed that the lead time demand follows a normal distribution and shortages are backordered. Also, buyer's lead time can be reduced at an added cost. A model has been formulated to find the optimal order quantity, lead time and the number of shipments from the vendor to the buyer during one production cycle while minimising the total expected cost per unit time of the vendor-buyer integrated system. It is often difficult to estimate the stock-out cost in inventory systems. Therefore, instead of having a stock-out term in the objective function, a service level constraint is included in the model. Due to mathematical complexities involved, it is difficult to establish the solutions analytically. An iterative procedure is developed to find the optimal solution using LINGO 10, and a numerical example is given to illustrate the results of the proposed model. Furthermore, the effects of key parameters are also studied to provide the insight into the structure of the solution. In addition, the savings of buyer and vendor are investigated from implementation of joint optimisation model over the model in which they minimise their own cost independently.     

Impacts of collaborative investment and inspection policies on the integrated inventory model with defective items

For an imperfect production system, to reduce quality-related costs, a manager may consider investing capital in quality improvement. In general, the investment expense in reducing the defective rate of items is often paid by the vendor. On the other hand, the buyer may inspect the product quality as the order is received which implies it incurs an inspection cost. In a supply chain integrated system, to accomplish global optimisation, the vendor and buyer can agree to jointly invest capital to improve the imperfect production processes, and the buyer can remove the inspection programme as the defective rate reaches a certain low-level. Hence, this paper investigates the impacts of collaborative investment and inspection policies on an integrated inventory model with defective items. The objective of this study is to seek the optimal order quantity, shipping times from the vendor to the buyer per production run, and the defective rate that minimise the joint total cost per unit time. An algorithm is developed to find the optimal solution. Several numerical examples are presented to demonstrate the proposed model and solution procedure, and then several management insights are obtained from the numerical examples.     

Flexible work-in-process production system in supply chain management under quality improvement

For a complex product production, any flexible manufacturing system with a work-in-process inventory is recommended for a supply chain management (SCM) system. Building a flexible manufacturing system increases the total cost of the supply chain; for this reason, a discrete investment is important. For flexible production systems, production rate within a finite specific interval of production rate as work-in-process inventory is calculated. The aim of the supply chain is to reduce the total cost when demand during the lead time is a random variable with a normal distribution. A crashing cost is utilised to reduce the duration of lead time within the supply chain system. A model is proposed to obtain the optimal flexible production rate with the reduced total cost of the supply chain. A classical optimisation technique is employed to obtain the closed-form and quasi-closed-form solutions of the decision variables. An improved algorithm is designed to obtain the global minimum cost of SCM under the framework of a flexible production system. An illustrative numerical example and sensitivity analysis are given to test the model. A numerical study proves that this model obtains the minimum cost with the optimal decision variables.     

Optimal integrated maintenance/production policy for randomly failing systems with variable failure rate

This article deals with the combined production and maintenance plans for a manufacturing system satisfying a random demand. We first establish an optimal production plan which minimises the average total inventory and production cost. Second, using this optimal production plan, and taking into account the deterioration of the machine according to its production rate, we derive an optimal maintenance schedule which minimises the maintenance cost. A numerical example illustrates the proposed approach, this analytical approach, based on a stochastic optimisation model and using the operational age concept, reveals the significant influence of the production rate on the deterioration of the manufacturing system and consequently on the integrated production/maintenance policy.     

Comparison of order-fulfilment performance in MTO and MTS systems with an inventory cost budget constraint

The make-to-order or make-to-stock decision is an important issue faced by firms in many industries. In the existing literature, optimisation models comparing the cost of making product to stock versus making it to order have been widely developed. Motivated by the problem faced by a machinery company, we examine the issue from a different perspective and formulate service-maximisation models with inventory cost budget constraints. We compare and analyse order-fulfilment performance measures for the two different production control systems. The goal is to identify the key influencing factors and devise a production strategy that maximises service performance subject to resource constraint. We show that the MTO production control system is preferred in the production system with low component values and long component processing times and high value and short lead time in the final assembly stage; while the MTS production control system is applicable to the production system with high component values and short component processing times and little value and long lead time in the final assembly stage.     

服务员休假的生产服务库存模型的稳态分析及最优生产策略

将服务员休假与生产服务库存相结合，研究带有服务员休假的 M/M/1 生产服务库存系统模型。需求 (顾客) 的到达过程服从 Poisson 过程，顾客的服务时间和每个产品的生产时间都服从指数分布。当系统库存为零时，服务员开始随机长度的休假。假设休假时间服从指数分布。首先，利用拟生灭过程理论给出了系统的稳态平衡条件。其次，对忽略服务时间的生产服务库存系统模型进行了稳态分析，得到了此系统的稳态概率的明显的解析表达式，进而证明了系统的稳态概率分布具有乘积解形式的结构。在此基础上，进一步得到了系统的一些稳态性能指标和费用函数的计算公式，数值求解了模型的最优 $(s,S)$ 库存策略，并研究系统的一些参数对性能指标、最优策略和最优费用的影响。     

Joint optimisation of maintenance and production policies considering random demand and variable production rate

This paper deals with a randomly failing manufacturing system M1 which has to satisfy a random demand during a finite horizon given a required service level. To help meet this demand, subcontracting is used through another production system M2. M1 operates with a variable production rate and its failure rate depends on both time and the production rate. In these conditions, as a first step, we establish a preliminary production plan corresponding to a given service level. In a second stage, we integrate the effect of the machine degradation introducing a unitary degradation cost. The optimal production plan is then obtained by minimising the sum of the production, the inventory and the degradation costs. In the final stage, we propose another optimal plan combined with a preventive maintenance policy aiming at reducing the machine degradation while minimising the total cost including the production, inventory and maintenance costs.     

Distributed algorithm for optimal sequence and power allocation in uplink code division multiple access systems

An alternative approach is presented for centralised algorithms that design optimal sequences and powers under quality of service constraints in the uplink of a code division multiple access systems. The authors propose a distributed algorithm, where each user designs its optimal codeword in such a way to transmit minimum power, based on certain feedback information sent from the base station. The authors define the user cost function, which is the user power written as a convex function of user codewords for a defined signal-to-interference plus noise ratio target. For the constrained optimisation problem, optimal codewords are designed based on feasible direction method and the optimal user powers are the minimum values of the user cost functions. For the optimal configuration, the matched filter employed at receiver will have the same performance as the minimum mean squared error filter. Even if the optimal user powers are unique, the optimal codewords do not correspond to a unique allocation, but rather to an entire class of codewords that can be related by unitary transformations. The algorithm works properly in the presence of multiple access interference with white or coloured additive noise and requires no ordering. The proposed algorithm is analysed and illustrated with numerical examples obtained from simulations.     

Dynamic and static job allocation for multi-server systems

We consider the optimal assignment of groups of jobs to a fixed number of time periods over a finite horizon to minimize the total facility idling and job waiting costs. The capacity of the facility varies randomly in the sense that the time that each one of the multiple servers becomes available is random (servers arrive late). The service times are also random and are independent and identically distributed. With approximations, we formulate a dynamic optimization model for this problem. With a simple modification, we can apply this dynamic model to a static outpatient appointment problem. We propose two methods to compute the capacity distribution: (1) Poisson approximation and (2) simulation. While the Poisson approximation works well for exponential service times, the simulation scheme enables us to use the dynamic model without actually specifying the service time distribution. The performance measures of the schedules obtained with these two methods compare well with those of the optimal allocation obtained from (exhaustive) simulation. We also conduct numerical studies to investigate the dynamics between the idling and waiting costs ratio and the number of scheduling periods.     

Tolerance optimization of a mobile phone camera lens system

In the manufacturing process for the lens system of a mobile phone camera, various types of assembly and manufacturing tolerances, such as tilt and decenter, should be appropriately allocated. Because these tolerances affect manufacturing cost and the expected optical performance, it is necessary to choose a systematic design methodology for determining optimal tolerances. In order to determine the tolerances that minimize production cost while satisfying the reliability constraints on important optical performance indices, we propose a tolerance design procedure for a lens system. A tolerance analysis is carried out using Latin hypercube sampling for evaluating the expected optical performance. The tolerance optimization is carried out using a function-based sequential approximate optimization technique that can reduce the computational burden and smooth numerical noise occurring in the optimization process. Using the proposed design approach, the optimal production cost was decreased by 28.3% compared to the initial cost while satisfying all the constraints on the expected optical performance. We believe that the tolerance analysis and design procedure presented in this study can be applied to the tolerance optimization of other systems.     

Optimisation of beam directions in intensity modulated radiation therapy planning

Abstract. In this paper we consider the problem of selecting optimal beam directions as well as optimal intensity profiles for radiation therapy. Our multiobjective mixed integer programming problem is based on and extends a multiobjective LP formulation for intensity optimisation by Hamacher and Küfer. We use a weighted sum scalarisation to explore the benefits of beam direction optimisation. We propose exact and heuristic methods for solving the problem and present some numerical results. Correspondence to: M. Ehrgott     

阈值调控的GI/Geo/1多重工作休假排队

在随机服务系统中,减少系统能源消耗,提高服务效率,是人们关注的焦点．基于此目的,本文加入阈值和休假中止策略,具体分析了离散时间GI/Geo/1工作休假排队系统．首先,建立了顾客到达前夕二维嵌入马尔可夫链,给出了GI/M/1型结构矩阵．其次,应用矩阵分析方法得到系统稳态队长分布,并进一步求得系统稳态下的平均队长和平均逗留时间等指标．最后,通过Matlab仿真软件,对模型中的性能指标给予刻画和展示,结果表明平均队长和平均逗留时间随阈值的增大而增大,随休假服务率的增加而减少．本文的结果将为交换虚通道以及无线网络等方面的研究提供一定的参考．