A continuous review inventory model with controllable backorder rate and investments

In this study, we investigate a continuous review inventory model to reduce lead time, yield variability and setup cost simultaneously through capital investments. We assume that the backorder rate is depending on the length of the lead time through the amount of shortages. We also assume that lead time demand's distribution is not known but its first and second moments are known. We apply minimax distribution free procedure to minimise the expected total annual cost. By using logarithmic investment function we describe the relationship between the reduction in lead time, yield variability and setup cost with capital investment. This function was used in many existing models. Our main aim is to determine the optimal capital investment and ordering policies that minimises the expected total annual cost. To find out the optimal solution, an algorithm is given. With the help of this algorithm, optimal capital investment and ordering policies are wrought out. Numerical examples are given to elucidate the model. Our proposed model greatly differs from the model existing in the literature (the model by Lin and Hou (2005 Lin, LC and Hou, KL. 2005. An Inventory System with Investment to Reduce Yield Variability and Setup Cost. Journal of the Operational Research Society, 56: 67–74. [Taylor &; Francis Online], [Web of Science ®] , [Google Scholar])) viz: (1) In the above model, yield variability and setup cost were reduced through capital investment. In our model we reduce yield variability setup cost and also the lead time, which plays a vital role in any business. By reducing lead time we can improve the service level to the customer so as to increase the competitive edge in business. (2) In the model (the model by Lin and Hou (2005 Lin, LC and Hou, KL. 2005. An Inventory System with Investment to Reduce Yield Variability and Setup Cost. Journal of the Operational Research Society, 56: 67–74. [Taylor &; Francis Online], [Web of Science ®] , [Google Scholar])), it was assumed that lead time demand follows normal distribution. But in our model we take the distribution of lead time demand as distribution free. That is, it can follow any distribution which is more general. (3) In the above model (the model by Lin and Hou (2005 Lin, LC and Hou, KL. 2005. An Inventory System with Investment to Reduce Yield Variability and Setup Cost. Journal of the Operational Research Society, 56: 67–74. [Taylor &; Francis Online], [Web of Science ®] , [Google Scholar])), shortages are completely backlogged. But we consider partial backlogging and take the backlogging rate as 0 ≤ B ≤ 1. If we set backlogging rate B = 1 we get the above model. That is, the above model is particular case of our model. (4) We also assume that the backorder rate depends on the length of the lead time through the amount of shortages. If the lead time is longer then shortage accumulation is higher. The patience of customers will result in failure in business since some customers may turn to some other supplier. Hence, the backorder rate will be reduced. This assumption is very realistic.     

Optimal control of the production–inventory system with deteriorating items and tradable emission permits

Using the well-known Arrow and Karlin (1958) Arrow, K.J., and Karlin, S. (1958), ‘Production over Time with Increasing Marginal Costs’, in Studies in the Mathematical Theory of Inventory and Production, eds. K.J. Arrow and S. Karlin, Stanford: Stanford University Press, pp. 61–69. [Google Scholar] dynamic production–inventory model and the model with tradable emission permits which was presented by Dobos (2005 Dobos, I. (2005), ‘The Effects of Emission Trading on Production and Inventories in the Arrow–Karlin Model’, International Journal of Production Economics, 93–94, 301–308.[Crossref], [Web of Science ®] , [Google Scholar], 2007) Dobos I. (2007), ‘Tradable Emission Permits and Production-inventory Strategies of the Firm’, International Journal of Production Economics, 108, 329–333.[Crossref], [Web of Science ®] , [Google Scholar], we develop a model of the production–inventory system with deteriorating items and tradable emission permits. The objective of this paper is to apply the optimal control theory to solve the production–inventory problem with deteriorating items and tradable emission permits, and derive the optimal inventory level and the optimal production rate that minimise the total cost. The results are discussed with a numerical example and a sensitivity analysis of the optimal solution with respect to the parameters of the production–inventory system is carried out.     

A note on ‘a replenishment policy for items with price-dependent demand,time-proportional deterioration and no shortages’

In a recent paper, Begum et al. (2012 Begum, R., Sahoo, R.R., and Sahu, S.K. (2012), ‘A Replenishment Policy for Items with Price-Dependent Demand, Time-Proportional Deterioration and no Shortages’, International Journal of Systems Science, 43, 903–910.[Taylor &; Francis Online], [Web of Science ®] , [Google Scholar], International Journal of Systems Science, 43, 903–910) established pricing and replenishment policy for an inventory system with price-sensitive demand rate, time-proportional deterioration rate which follows three parameters, Weibull distribution and no shortages. In their model formulation, it is observed that the retailer's stock level reaches zero before the deterioration occurs. Consequently, the model resulted in traditional inventory model with price sensitive demand rate and no shortages. Hence, the main purpose of this note is to modify and present complete model formulation for Begum et al. (2012) Begum, R., Sahoo, R.R., and Sahu, S.K. (2012), ‘A Replenishment Policy for Items with Price-Dependent Demand, Time-Proportional Deterioration and no Shortages’, International Journal of Systems Science, 43, 903–910.[Taylor &; Francis Online], [Web of Science ®] , [Google Scholar]. The proposed model is validated by a numerical example and the sensitivity analysis of parameters is carried out.     

Statistical inference for the lifetime performance index based on generalised order statistics from exponential distribution

In manufacturing industries, the lifetime of an item is usually characterised by a random variable X and considered to be satisfactory if X exceeds a given lower lifetime limit L. The probability of a satisfactory item is then η_L := P(X ≥ L), called conforming rate. In industrial companies, however, the lifetime performance index, proposed by Montgomery and denoted by C_L, is widely used as a process capability index instead of the conforming rate. Assuming a parametric model for the random variable X, we show that there is a connection between the conforming rate and the lifetime performance index. Consequently, the statistical inferences about η_L and C_L are equivalent. Hence, we restrict ourselves to statistical inference for C_L based on generalised order statistics, which contains several ordered data models such as usual order statistics, progressively Type-II censored data and records. Various point and interval estimators for the parameter C_L are obtained and optimal critical regions for the hypothesis testing problems concerning C_L are proposed. Finally, two real data-sets on the lifetimes of insulating fluid and ball bearings, due to Nelson (1982) Nelson, W. (1982), Applied Life Data Analysis, New York: John Wiley &; Sons.[Crossref] , [Google Scholar] and Caroni (2002) Caroni, C. (2002), ‘The correct “ball bearings” data’, Lifetime Data Analysis, 8, 395–399.[Crossref], [PubMed], [Web of Science ®] , [Google Scholar], respectively, and a simulated sample are analysed.     

Comment on ‘Persistent inputs and the standard H2-multivariable control problem’ by K. Park and J.J. Bongiorno Jr

It is necessary to welcome the publication of the paper by Park and Bongiorno Jr (Park, K. and Bongiorno Jr, J.J. (2009 Park, K and Bongiorno Jr, JJ. 2009. Persistent Inputs and the Standard H₂-multivariable Control Problem. International Journal of Control, 82: 2002–2012. [Taylor &; Francis Online], [Web of Science ®] , [Google Scholar]), ‘Persistent Inputs and the Standard H₂-multivariable Control Problem’, International Journal of Control, 82, 2002–2012) in which the special cases in H₂-optimisation problems are considered. However, for correct orientation of readers we would like to note some publications which, in our opinion, are connected to a problem given by Park and Bongiorno Jr (2009 Park, K and Bongiorno Jr, JJ. 2009. Persistent Inputs and the Standard H₂-multivariable Control Problem. International Journal of Control, 82: 2002–2012. [Taylor &; Francis Online], [Web of Science ®] , [Google Scholar]), but are absent there.     

Further results on sampled-data design for robust control of a single qubit

This paper presents further results on the robust control method for qubit systems in Dong et al. (2013) Dong, D., Petersen, I.R., &; Rabitz, H. (2013). Sampled-data design for robust control of a single qubit. IEEE Transactions on Automatic Control, 58, 2654–2659.[Crossref], [Web of Science ®] , [Google Scholar]. Based on the properties of an antisymmetric system, an alternative method is presented to analyse and exclude singularity intervals in the proof of partial original results. For the case of amplitude damping decoherence, a larger sampling period is presented when the upper bound of the probability of failure is small enough. For the case of phase damping decoherence, a larger sampling period is given when the lower bound of the target coherence is large enough. Furthermore, we provide improved sampling periods for amplitude damping decoherence and phase damping decoherence without the above prior constraints.     

Backstepping-based cooperative and adaptive tracking control design for a group of underactuated AUVs in horizontal plan

In this paper, we investigate new implementable cooperative adaptive backstepping controllers for a group of underactuated autonomous vehicles that are communicating with their local neighbours to track a time-varying virtual leader of which the relative position may only be available to a portion of the team members. At the kinematic cooperative control level of the autonomous underwater vehicle, the virtual cooperative controller is basically designed on a proportional and derivative consensus algorithm presented in Ren (2010) Ren, W. (2010). Consensus tracking under directed interaction topologies: Algorithms and experiments. IEEE Transactions on Control Systems Technology, 18, 230–237.[Crossref], [Web of Science ®] , [Google Scholar], which involves velocity information from local neighbours. In this paper, we propose a new design algorithm based on singular perturbation theory that precludes the use of the neighbours’ velocity information in the cooperative design. At the dynamic cooperative control level, calculation of the partial derivatives of some stabilising functions which in turn will contain velocity information from the local neighbours is required. To facilitate the implementation of the cooperative controllers, we propose a command filter approach technique to avoid analytic differentiation of the virtual cooperative control laws. We show how Lyapunov-based techniques and graph theory can be combined together to yield a robust cooperative controller where the uncertain dynamics of the cooperating vehicles and the constraints on the communication topology which contains a directed spanning tree are explicitly taken into account. Simulation results with a dynamic model of underactuated autonomous underwater vehicles moving on the horizontal plane are presented and discussed.     

The effects of individual differences,prior experience and cognitive load on the transfer of dynamic decision-making performance

Situational awareness is recognised as an important factor in the performance of individuals and teams in dynamic decision-making (DDM) environments (Salmon et al. 2014 Salmon, P. M., N. A.Stanton, G. H.Walker, D.Jenkins, D.Ladva, L.Rafferty, and M.Young. 2014. “Measuring SA in Complex Systems: Comparison of Measurement Study.” International Journal of Industrial Ergonomics39: 490–500.[Crossref], [Web of Science ®] , [Google Scholar]). The present study was designed to investigate whether the scores on the WOMBAT? Situational Awareness and Stress Tolerance Test (Roscoe and North 1980 Roscoe, S. N., and R.North. 1980. “Prediction of Pilot Performance.” In Aviation Psychology, edited by S. N.Roscoe, 123–127. Ames: The Iowa State University Press. [Google Scholar]) would predict the transfer of DDM performance from training under different levels of cognitive load to a novel situation. Participants practised a simulated firefighting task under either low or high conditions of cognitive load and then performed a (transfer) test in an alternative firefighting environment under an intermediate level of cognitive load. WOMBAT? test scores were a better predictor of DDM performance than scores on the Raven Matrices. Participants with high WOMBAT? scores performed better regardless of their training condition. Participants with recent gaming experience who practised under low cognitive load showed better practice phase performance but worse transfer performance than those who practised under high cognitive load.

Practitioner Summary: The relationship between task experience, situational awareness ability, cognitive load and the transfer of dynamic decision-making (DDM) performance was investigated. Results showed that the WOMBAT? test predicted transfer of DDM performance regardless of task cognitive load. The effects of cognitive load on performance varied according to previous task-relevant experience.     

Comment on ‘Optimal inventory replenishment policy for the EPQ model under trade credit derived without derivatives’

In this article, we complement the shortcoming of the inventory economic production quantity (EPQ) model developed by Huang and Huang (2008 Huang, YF and Huang, HF. 2008. Optimal Inventory Replenishment Policy for the EPQ Model Under Trade Credit Derived Without Derivatives. International Journal of Systems Science, 39: 539–546. [Taylor &; Francis Online], [Web of Science ®] , [Google Scholar]), ‘Optimal Inventory Replenishment Policy for the EPQ Model Under Trade Credit Derived Without Derivatives’, International Journal of Systems Science, 39, 539–546), and propose an arithmetic–geometric inequality method to obtain the global optimal solution without taking complex differential calculus or using tedious algebraic manipulations. Finally, we provide an economical interpretation of the theoretical result so that the reader can understand the insight of the result.     

Continuous-time non-linear flatness-based predictive control: an exact feedforward linearisation setting with an induction drive example

A general flatness-based framework for non-linear continuous-time predictive control is presented. It extends the results of Fliess and Marquez (2000 Fliess, M and Marquez, R. 2000. Continuous-time Linear Predictive Control and Flatness: A Module-theoretic Setting with Examples. International Journal of Control, 73: 606–623. [Taylor &; Francis Online], [Web of Science ®] , [Google Scholar]) to the non-linear case. The mathematical setting, which is valid for multivariable systems, is provided by the theory of flatness-based exact feedforward linearisation introduced by the authors (Hagenmeyer and Delaleau 2003b Hagenmeyer, V and Delaleau, E. 2003b. Exact Feedforward Linearization Based on Differential Flatness. International Journal of Control, 76: 537–556. [Taylor &; Francis Online], [Web of Science ®] , [Google Scholar]). Thereby differential flatness does not only yield an easy calculation of the predicted trajectories considering the respective system constraints, but allows to use simple linear feedback parts in a two-degree-of-freedom control structure. Moreover, this formalism permits one to handle non-minimum phase systems, and furthermore to deal with parameter uncertainties and exogenous perturbations. Respective robustness analysis tools are available. Finally, an induction drive example is discussed in detail and experimental results for this fast electro-mechanical system are presented.     

Approximation error in PDE-based modelling of vehicular platoons

We study the problem of how much error is introduced in approximating the dynamics of a large vehicular platoon by using a partial differential equation, as was done in Barooah, Mehta, and Hespanha [Barooah, P., Mehta, P.G., and Hespanha, J.P. (2009 Barooah, P, Mehta, PG and Hespanha, JP. 2009. Mistuning-based Decentralised Control of Vehicular Platoons for Improved Closed Loop Stability. IEEE Transactions on Automatic Control, 54: 2100–2113. [Crossref], [Web of Science ®] , [Google Scholar]), ‘Mistuning-based Decentralised Control of Vehicular Platoons for Improved Closed Loop Stability’, IEEE Transactions on Automatic Control, 54, 2100–2113], Hao, Barooah, and Mehta [Hao, H., Barooah, P., and Mehta, P.G. (2011), ‘Stability Margin Scaling Laws of Distributed Formation Control as a Function of Network Structure’, IEEE Transactions on Automatic Control, 56, 923–929]. In particular, we examine the difference between the stability margins of the coupled-ordinary differential equations (ODE) model and its partial differential equation (PDE) approximation, which we call the approximation error. The stability margin is defined as the absolute value of the real part of the least stable pole. The PDE model has proved useful in the design of distributed control schemes (Barooah et al. 2009 Barooah, P, Mehta, PG and Hespanha, JP. 2009. Mistuning-based Decentralised Control of Vehicular Platoons for Improved Closed Loop Stability. IEEE Transactions on Automatic Control, 54: 2100–2113. [Crossref], [Web of Science ®] , [Google Scholar]; Hao et al. 2011 Hao, H, Barooah, P and Mehta, PG. 2011. Stability Margin Scaling Laws of Distributed Formation Control as a Function of Network Structure. IEEE Transactions on Automatic Control, 56: 923–929. [Crossref], [Web of Science ®] , [Google Scholar]); it provides insight into the effect of gains of local controllers on the closed-loop stability margin that is lacking in the coupled-ODE model. Here we show that the ratio of the approximation error to the stability margin is O(1/N), where N is the number of vehicles. Thus, the PDE model is an accurate approximation of the coupled-ODE model when N is large. Numerical computations are provided to corroborate the analysis.     

FROM THE ARCHIVES: CODETALKERS NOT WANTED

Abstract

The Hill Cipher, also known as matrix encryption, uses matrices to encipher and decipher text. Various attacks, such as those found by Jack Levine [2 Levine , J. 1961 . “Some Applications of High-Speed Computers to the Case n = 2 of Algebraic Cryptography,” Mathematics of Computation , 15 ( 75 ). [Google Scholar] 3 Levine , J. 1961 . “Some Elementary Cryptanalysis of Algebraic Cryptography,” American Mathematical Monthly , 68 ( 5 ): 411 – 418 .[Taylor &; Francis Online] , [Google Scholar] 5 Levine , J. and R. Chandler . 1989 . “The Hill Cryptographic System with Unknown Cipher Alphabet but Known Plaintext,” Cryptologia , 13 ( 1 ): 1 – 28 .[Taylor &; Francis Online] , [Google Scholar]], have been published for this system. This article reviews a few previous results and presents a powerful new attack in which the rows of the matrix can be determined independent of one another, greatly reducing the amount of time needed for decipherment.     

Decryption oracle slide attacks on T-310

T-310 is an important Cold War cipher (Schmeh 2006 Schmeh, K. 2006. The East German encryption machine T-310 and the algorithm it used. Cryptologia 30 (3):251–257.[Taylor &; Francis Online] , [Google Scholar]). It was the principal encryption algorithm used to protect various state communication lines in Eastern Germany throughout the 1980s. The cipher seems to be quite robust, and until now no researcher has proposed an attack on T-310. This article studies decryption oracle and slide attacks on T-310.     

A parallel time-memory tradeoff attack on the Hill cipher

Leap et al. (2016 Leap, T., T. McDevitt, K. Novak, and N. Siermine. 2016. Further improvements to the Bauer-Millward attack on the Hill cipher. Cryptologia 40:1–16.[Taylor &; Francis Online], [Web of Science ®] , [Google Scholar]) reduced the time complexity of the Bauer-Millward (2007 Bauer, C., and K. Millward. 2007. Cracking matrix encryption row by row. Cryptologia 31(1):76–83.[Taylor &; Francis Online], [Web of Science ®] , [Google Scholar]) ciphertext-only attack on the Hill cipher from 𝒪(Lⁿ) to 𝒪(L^n?1), where L is the length of the alphabet, and n is the block size. This article presents an attack that reduces the complexity to 𝒪(L^n?1?s), 0?≤?s?≤?n???1. The practical limitation on the size of s is the memory available on the computer being used for the attack. Specifically, the computer must be able to hold L^s integer arrays of length N, where N is the number of blocks of ciphertext. The key idea is not to iterate over potential rows of the decryption matrix, but to iterate over randomly chosen characters in the plaintext. This attack also admits a straightforward parallel implementation on multiple processors to further decrease the run time of the attack.     

A numerical algorithm to find soft-constrained Nash equilibria in scalar LQ-games

This paper provides a numerical algorithm to calculate all soft-constrained Nash equilibria in a regular scalar indefinite linear-quadratic game. The algorithm is based on the calculation of the eigenstructure of a certain matrix. The analysis follows the lines of the approach taken by Engwerda (2003 Engwerda, JC. 2003. Solving the scalar feedback Nash algebraic Riccati equations: an eigenvector approach. IEEE Trans. Automat. Contr., 48: 847–853. [Crossref] , [Google Scholar]) to calculate the solutions of a set of scalar coupled feedback Nash algebraic Riccati equations.     

Capture of shrinking targets

‘Shrinking targets’ are targets whose size diminishes with time. The task studied is a modification of Fitts' (1954 Fitts, P. M. 1954. The information capacity of the human motor system in controlling the amplitude of movement. Journal of Experimental Psychology, 67(6): 381–391. [Crossref] , [Google Scholar]) paradigm, with the difference that, as soon as the movement is started, the target size reduces at a constant rate until it finally vanishes. Very little research has been reported on this problem apart from Johnson and Hart (1987 Johnson, W. W. and Hart, S. G. Step tracking shrinking targets. Proceedings of the Human Factors Society 31st annual meeting. October1987, New York City. pp.248–252. Santa Monica, CA: HFES.  [Google Scholar]) and Hancock and Caird (1993 Hancock, P. A. and Caird, J. K. 1993. Experimental evaluation of a model of mental workload. Human Factors, 35(3): 413–429. [Crossref], [PubMed], [Web of Science ®] , [Google Scholar]). Two experiments are reported aimed at determining the parameters that affect the movement time and the probability of capturing a target when there are different amplitudes of movement, target widths and shrink rates. A multiplicative model is required to describe movement time data, which is dependent on Fitts' Index of Difficulty, the shrink rate and the product of these two variables. An alternative model describes the critical movement time, for a specified probability of target capture, in a modified form of Fitts' Law.

Statement of Relevance: Modifications of Fitts' Law have been developed for many different movement tasks. Shrinking targets occur in circumstances such as gunnery and in computer games, where a target is moving away from the person. An expression is developed for the critical time to capture the target in terms of a modified form of Fitts' Law.