Task assignment under uncertainty: stochastic programming and robust optimisation approaches

The assignment of tasks to teams is a challenging combinatorial optimisation problem. The uncertainty in the tasks’ execution processes further complicates the assignment decisions. This study investigates a variant of the typical assignment problem, in which each task can be divided into two parts, one is deterministic and the other is uncertain with respect to their workloads. From the stochastic perspective, this paper proposes both a stochastic programming model that can cope with arbitrary probability distributions of tasks’ random workload requirements, and a robust optimisation model that is applicable to situations in which limited information about probability distributions is available. An example of its application in the software project management is given. Some numerical experiments are also performed to validate the effectiveness of the proposed models and the relationships between the two models.     

Integrated multi-objective robust optimization and sensitivity analysis with irreducible and reducible interval uncertainty

Uncertainty considered in robust optimization is usually treated as irreducible since it is not reduced during an optimization procedure. In contrast, uncertainty considered in sensitivity analysis is treated as partially or fully reducible in order to quantify the effect of input uncertainty on the outputs of the system. Considering this, and the usual existence of both reducible and irreducible uncertainty, an approach that can perform robust optimization and sensitivity analysis simultaneously is of much interest. This article presents such an integrated optimization model that can be used for both robust optimization and sensitivity analysis for problems that have irreducible and reducible interval uncertainty, multiple objective functions and mixed continuous-discrete design variables. The proposed model is demonstrated by two engineering examples with differing complexity to demonstrate its applicability.     

Fault-tolerant embedded system design and optimization considering reliability estimation uncertainty

In this paper, we model embedded system design and optimization, considering component redundancy and uncertainty in the component reliability estimates. The systems being studied consist of software embedded in associated hardware components. Very often, component reliability values are not known exactly. Therefore, for reliability analysis studies and system optimization, it is meaningful to consider component reliability estimates as random variables with associated estimation uncertainty. In this new research, the system design process is formulated as a multiple-objective optimization problem to maximize an estimate of system reliability, and also, to minimize the variance of the reliability estimate. The two objectives are combined by penalizing the variance for prospective solutions. The two most common fault-tolerant embedded system architectures, N-Version Programming and Recovery Block, are considered as strategies to improve system reliability by providing system redundancy. Four distinct models are presented to demonstrate the proposed optimization techniques with or without redundancy. For many design problems, multiple functionally equivalent software versions have failure correlation even if they have been independently developed. The failure correlation may result from faults in the software specification, faults from a voting algorithm, and/or related faults from any two software versions. Our approach considers this correlation in formulating practical optimization models. Genetic algorithms with a dynamic penalty function are applied in solving this optimization problem, and reasonable and interesting results are obtained and discussed.     

A robust optimisation model for production planning and pricing under demand uncertainty

The profitability of every manufacturing plant is dependent on its pricing strategy and a production plan to support the customers’ demand. In this paper, a new robust multi-product and multi-period model for planning and pricing is proposed. The demand is considered to be uncertain and price-dependent. Thus, for each price, a range of demands is possible. The unsatisfied demand is considered to be lost and hence, no backlogging is allowed. The objective is to maximise the profit over the planning horizon, which consists of a finite number of periods. To solve the proposed model, a modified unconscious search (US) algorithm is introduced. Several artificial test problems along with a real case implementation of the model in a textile manufacturing plant are used to show the applicability of the model and effectiveness of the US for tackling this problem. The results show that the proposed model can improve the profitability of the plant and the US is able to find high quality solutions in a very short time compared to exact methods.     

Robust optimization based on analytical evaluation of uncertainty propagation

Optimization under uncertainty requires proper handling of those input parameters that contain scatter. Scatter in input parameters propagates through the process and causes scatter in the output. Stochastic methods (e.g. Monte Carlo) are very popular for assessing uncertainty propagation using black-box function metamodels. However, they are expensive. Therefore, in this article a direct method of calculating uncertainty propagation has been employed based on the analytical integration of a metamodel of a process. Analytical handling of noise variables not only improves the accuracy of the results but also provides the gradients of the output with respect to input variables. This is advantageous in the case of gradient-based optimization. Additionally, it is shown that the analytical approach can be applied during sequential improvement of the metamodel to obtain a more accurate representative model of the black-box function and to enhance the search for the robust optimum.     

A bi-objective robust model for emergency resource allocation under uncertainty

Emergency resource allocation constitutes one of the most critical elements of response operations in the field of emergency management. This paper addresses an emergency resource allocation problem which involves multiple competing affected areas and one relief resource centre under supply shortage and uncertainty in the post-disaster phase. In humanitarian situations, both the efficiency and fairness of an allocation policy have a considerable influence on the effectiveness of emergency response operations. Thus, we formulate a bi-objective robust emergency resource allocation (BRERA) model which tries to maximise efficiency as well as fairness under different sources of uncertainties. To obtain decision-makers’ most preferred allocation policy, we propose a novel emergency resource allocation decision method which consists of three steps: (1) develop a bi-objective heuristic particle swarm optimisation algorithm to search the Pareto frontier of the BRERA model; (2) select a coefficient to measure fairness; and (3) establish a decision method based on decision-makers’ preference restricted by the fairness coefficient. Finally, a real case study taken from the 5 December 2008 Wenchuan Earthquake demonstrates the effectiveness of the proposed method through numerical results. The solution and model robustness are also analysed.     

A single-loop optimization method for reliability analysis with second order uncertainty

Reliability analysis may involve random variables and interval variables. In addition, some of the random variables may have interval distribution parameters owing to limited information. This kind of uncertainty is called second order uncertainty. This article develops an efficient reliability method for problems involving the three aforementioned types of uncertain input variables. The analysis produces the maximum and minimum reliability and is computationally demanding because two loops are needed: a reliability analysis loop with respect to random variables and an interval analysis loop for extreme responses with respect to interval variables. The first order reliability method and nonlinear optimization are used for the two loops, respectively. For computational efficiency, the two loops are combined into a single loop by treating the Karush–Kuhn–Tucker (KKT) optimal conditions of the interval analysis as constraints. Three examples are presented to demonstrate the proposed method.     

Collaborative optimization with inverse reliability for multidisciplinary systems uncertainty analysis

This article introduces a method which combines the collaborative optimization framework and the inverse reliability strategy to assess the uncertainty encountered in the multidisciplinary design process. This method conducts the sub-system analysis and optimization concurrently and then improves the process of searching for the most probable point (MPP). It reduces the load of the system-level optimizer significantly. This advantage is specifically more prominent for large-scale engineering system design. Meanwhile, because the disciplinary analyses are treated as the equality constraints in the disciplinary optimization, the computation load can be further reduced. Examples are used to illustrate the accuracy and efficiency of the proposed method.     

Robust topology optimization under load position uncertainty

The topology optimization problem of a continuum structure on the compliance minimization objective is investigated under consideration of the external load uncertainty in its application position with a nonprobabilistic approach. The load position is defined as the uncertain-but-bounded parameter and is represented by an interval variable with a nominal application point. The structural compliance due to the load position deviation is formulated with the quadratic Taylor series expansion. As a result, the objective gradient information to the topological variables can be evaluated efficiently in a quadratic expression. Based on the maximum design sensitivity value, which corresponds to the most sensitive compliance to the uncertain loading position, a single-level optimization approach is suggested by using a popular gradient-based optimality criteria method. The proposed optimization scheme is performed to gain the robust topology optimizations of three benchmark examples, and the final configuration designs are compared comprehensively with the conventional topology optimizations under the loading point fixation. It can be observed that the present method can provide remarkably different material layouts with auxiliary components to accommodate the load position disturbances. The numerical results of the representative examples also show that the structural performances of the robust topology optimizations appear less sensitive to the load position perturbations than the traditional designs.     

Demonstration of probabilistic ordinal optimization concepts for continuous-variable optimization under uncertainty

A very general and robust approach to solving optimization problems involving probabilistic uncertainty is through the use of Probabilistic Ordinal Optimization. At each step in the optimization problem, improvement is based only on a relative ranking of the probabilistic merits of local design alternatives, rather than on precise quantification of the alternatives. Thus, we simply ask the question: “Is that alternative better or worse than this one?” to some level of statistical confidence we require, not: “HOW MUCH better or worse is that alternative to this one?”. In this paper we illustrate an elementary application of probabilistic ordinal concepts in a 2-D optimization problem. Two uncertain variables contribute to uncertainty in the response function. We use a simple Coordinate Pattern Search non-gradient-based optimizer to step toward the statistical optimum in the design space. We also discuss more sophisticated implementations, and some of the advantages and disadvantages versus other approaches to optimization under uncertainty.     

An optimization model for a crop deficit irrigation system under uncertainty

In this study, an inexact nonlinear programming model under uncertainty is developed by incorporating a water production function into the crop irrigation system optimization framework. By introducing a time parameter, this model can address the uncertainty associated with the irrigation schedule for different crops and their planting stages. The developed model was applied to a case study of an agricultural water resources management problem to demonstrate its applicability. Through scenario analysis under different precipitation levels, the key planting stage of crops and the amount of water for the irrigation schedule that could significantly affect system benefits were identified. By using intervals to represent uncertain parameters, more reliable and practical decision alternatives were generated through the presented model in typical hydrological years (i.e. wet, normal and dry years).     

双渠道供应链的库存与定价策略研究

针对由传统零售渠道和网络直销渠道组成的双渠道供应链模型,考虑批发价格和传统零售价格确定情况下,同时市场需求随机且受价格影响时,对制造商在直销渠道上的最优定价和库存量决策,以及零售商在传统分销渠道上的最优订货量进行研究。通过模型分析和数值仿真说明：当需求分配比例处在一定范围内时,存在可行的直销价格。当加入直销价格不小于批发价格的约束条件后,在更小的需求分配比例范围内能够找到均衡解。且在此合理的区域内,传统零售渠道订货量减少,直销渠道库存量增大,但供应链总的销售量基本不变。零售商的期望收益有所增加,对于供应商,其来自于零售渠道的期望收益减少,来自于直销渠道的收益以及总收益增多。整个供应链的收益得到提高。     

Project management for uncertainty with multiple objectives optimisation of time,cost and reliability

This research adopts an approach that uses computer simulation and statistical analysis of uncertain activity time, activity cost, due date and project budget to address quality and the learning process with regard to project scheduling. Since the learning process affects the scheduling problem, a Cobb–Douglas multiplicative power model is used to represent the relationship between the dependent variable, which is the standard deviation of activity time, and the independent variables, which are the cumulative trials and the mean of activity time. The mean value and standard deviation are used to randomly generate activity times for project scheduling analysis. Response surface methodology (RSM) is used in order to develop a rationale of the time-cost trade-off problem. The solutions found with RSM are optimised only for a single objective, such as project completion time, total project cost, completion time probability and total cost probability. Thus, multiple objectives for further optimisation become necessary and a limited project budget, restricted completion time, allowable total cost probability and acceptable completion time probability have to be considered at the same time as the learning effect. With response functions from RSM, compromise programming is adopted in order to formulate the proposed project scheduling problem for multi-objective optimisation.     

Source/relays/destination combined MMSE-based optimization for AF-MIMO multiple-relay systems with correlated channel uncertainties

In this paper, source/relays-precoders and destination-equalizer combined optimization are proposed as a dual-hop amplify-and-forward (AF) multiple-input multiple-output (MIMO) multiple-relay system with Gaussian random and correlated channel uncertainties in both hops. Taking correlated channel uncertainties into account, a robust transceiver joint optimization design is developed based on the minimum mean-squared error (MMSE) criterion under individual power constraints at the source and the relays. Simulation results illustrate that the robust multiple relays/transceiver joint design architecture for an AF-MIMO system equipped with multiple relays substantially outperforms a nonrobust transceiver design that assumes estimated channels as actual channels.     

Game formulations for structural optimization under uncertainty

We consider structural optimization (SO) under uncertainty formulated as a mathematical game between two players –– a “designer” and “nature”. The first player wants to design a structure that performs optimally, whereas the second player tries to find the worst possible conditions to impose on the structure. Several solution concepts exist for such games, including Stackelberg and Nash equilibria and Pareto optima. Pareto optimality is shown not to be a useful solution concept. Stackelberg and Nash games are, however, both of potential interest, but these concepts are hardly ever discussed in the literature on SO under uncertainty. Based on concrete examples of topology optimization of trusses and finite element-discretized continua under worst-case load uncertainty, we therefore analyze and compare the two solution concepts. In all examples, Stackelberg equilibria exist and can be found numerically, but for some cases we demonstrate nonexistence of Nash equilibria. This motivates a view of the Stackelberg solution concept as the correct one. However, we also demonstrate that existing Nash equilibria can be found using a simple so-called decomposition algorithm, which could be of interest for other instances of SO under uncertainty, where it is difficult to find a numerically efficient Stackelberg formulation.     

多品牌商线上线下竞争定价与渠道决策研究

为解决多个品牌商线上线下渠道决策问题,运用Spokes模型分析了直营和混合销售模式的最优定价及收益情况,基于博弈论方法,研究了市场和渠道竞争程度对品牌商决策均衡的影响。结果表明：两种销售模式下,品牌商的最优定价随市场和渠道竞争程度的增加而增加;当市场或渠道竞争程度较低时,品牌商选择直营模式;市场或渠道竞争程度处于一定阈值区间时,两种模式都具有适用性;而市场或渠道竞争程度较高则选择混合模式。     

考虑渠道成本差异的双渠道供应链定价策略与渠道选择

根据网上渠道不同销售模式的特征,构建制造商网上直销、网上分销和网上代销3种供应链定价模型,引入渠道成本参数并分析制造商如何进行产品定价和渠道选择。研究表明,在3种渠道结构中,代销模式下产品价格最低,直销模式和分销模式下的价格受到渠道偏好的影响;随着线上或线下渠道成本增加,制造商会降低该渠道产品的批发价格,而零售价格与自身渠道成本正相关;此外,当制造商自建网上平台成本较低时,制造商选择直销模式最优;当制造商自建网上平台成本较高,且代销平台服务效率较低时,制造商应选择分销模式,否则制造商选择代销模式更优。     

Multifidelity approaches for optimization under uncertainty

It is important to design robust and reliable systems by accounting for uncertainty and variability in the design process. However, performing optimization in this setting can be computationally expensive, requiring many evaluations of the numerical model to compute statistics of the system performance at every optimization iteration. This paper proposes a multifidelity approach to optimization under uncertainty that makes use of inexpensive, low‐fidelity models to provide approximate information about the expensive, high‐fidelity model. The multifidelity estimator is developed based on the control variate method to reduce the computational cost of achieving a specified mean square error in the statistic estimate. The method optimally allocates the computational load between the two models based on their relative evaluation cost and the strength of the correlation between them. This paper also develops an information reuse estimator that exploits the autocorrelation structure of the high‐fidelity model in the design space to reduce the cost of repeatedly estimating statistics during the course of optimization. Finally, a combined estimator incorporates the features of both the multifidelity estimator and the information reuse estimator. The methods demonstrate 90% computational savings in an acoustic horn robust optimization example and practical design turnaround time in a robust wing optimization problem. Copyright © 2014 John Wiley & Sons, Ltd.     

Dual-channel closed loop supply chains: forward channel competition,power structures and coordination

This paper examines the effect of forward channel competition and power structure on dual-channel closed loop supply chains (CLSC), which consists of a manufacturer, a retailer and a collector. The manufacturer can either wholesale products to the retailer or directly sell them to the market, the collector undertakes the collection activity of used products. Under different channel power structures, a centralised and three decentralised models are explored under symmetric and asymmetric relative status between direct and retail channels. Through a comprehensive comparison among these models, the result shows that each channel member has an incentive to play the channel leader’s role. Meanwhile, from the total channel system’s perspective, we find that the manufacturer-led or retailer-led model can either be the most effective CLSC under symmetric relative channel status, which depends on the channel substitution rate between two channels. While under asymmetric relative channel status, the numerical result shows that the whole CLSC should select collector-led, manufacturer-led and retailer-led CLSC model in turn with the retail channel’s relative status becoming stronger. Finally, with the benchmark of the centralised decision-making system, we design modified two-part tariff contracts to coordinate dual-channel CLSCs under different channel power structures.     

TQM and environmental uncertainty levels: profiles,fit, and firm performance

That a manufacturer should align its implementation of Total Quality Management (TQM) to the external environment it faces has been indirectly argued for long. Theoretical and empirical evidence for this argument has, unfortunately, been lacking. Our research remedies this knowledge gap. Borrowing structural contingency theory and the concept of fit, we hypothesised and report three findings. First, we found contrasting normative TQM profiles between a high vs. a low level of environmental uncertainty. Second, we identified significant detrimental impact on firm performance when a firm deviates its TQM implementation from the normative TQM profile prescribed for a specific level of environmental uncertainty. Third, we also discovered discernible differences in the deterioration in firm performance between a positive vs. a negative deviation from the normative TQM profile. These robust findings were derived from analysing secondary survey data from 330 Chinese manufacturing firms via profile deviation analysis, MANOVA, MANCOVA and OLS regression. Contrary to the literature, manufacturers operating in a volatile external environment should pursue and benefit from TQM implementation. Manufacturers should, however, not seek to implement TQM to the fullest extent nor implement TQM half-heartedly. Instead, manufacturers should benchmark best performers as to what the normative TQM profile is and pursue their own TQM implementation to minimise deviations from the normative TQM profile.