A new approach towards integrated cell formation and inventory lot sizing in an unreliable cellular manufacturing system

This paper presents a comprehensive mathematical model for integrated cell formation and inventory lot sizing problem. The proposed model seeks to minimize cell formation costs as well as the costs associated with production, while dynamic conditions, alternative routings, machine capacity limitation, operations sequences, cell size constraints, process deterioration, and machine breakdowns are also taken into account. The total cost consists of machine procurement, cell reconfiguration, preventive and corrective repairs, material handling (intra-cell and inter-cell), machine operation, part subcontracting, finished and unfinished parts inventory cost, and defective parts replacement costs. With respect to the multiple products, multiple process plans for each product and multiple routing alternatives for each process plan which are assumed in the proposed model, the model is combinatorial. Moreover, unreliability conditions are considered, because moving from “in-control” state to “out-of-control” state (process deterioration) and machine breakdowns make the model more practical and applicable. To conquer the breakdowns, preventive and corrective actions are adopted. Finally, a Particle Swarm Optimization (PSO)-based meta-heuristic is developed to overcome NP-completeness of the proposed model.     

An optimal replacement policy for a multistate degenerative simple system

In this paper, a degenerative simple system (i.e. a degenerative one-component system with one repairman) with k + 1 states, including k failure states and one working state, is studied. Assume that the system after repair is not “as good as new”, and the degeneration of the system is stochastic. Under these assumptions, we consider a new replacement policy T based on the system age. Our problem is to determine an optimal replacement policy T^∗ such that the average cost rate (i.e. the long-run average cost per unit time) of the system is minimized. The explicit expression of the average cost rate is derived, the corresponding optimal replacement policy can be determined, the explicit expression of the minimum of the average cost rate can be found and under some mild conditions the existence and uniqueness of the optimal policy T^∗ can be proved, too. Further, we can show that the repair model for the multistate system in this paper forms a general monotone process repair model which includes the geometric process repair model as a special case. We can also show that the repair model in the paper is equivalent to a geometric process repair model for a two-state degenerative simple system in the sense that they have the same average cost rate and the same optimal policy. Finally, a numerical example is given to illustrate the theoretical results of this model.     

An extended replacement policy for a deteriorating system with multi-failure modes

In this paper, the maintenance problem for a deteriorating system with k + 1 failure modes, including an unrepairable failure (catastrophic failure) mode and k repairable failure (non-catastrophic failure) modes, is studied. Assume that the system after repair is not “as good as new” and its deterioration is stochastic. Under these assumptions, an extended replacement policy N is considered: the system will be replaced whenever the number of repairable failures reaches N or the unrepairable failure occurs, whichever occurs first. Our purpose is to determine an optimal extended policy N^∗ such that the average cost rate (i.e. the long-run average cost per unit time) of the system is minimized. The explicit expression of the average cost rate is derived, and the corresponding optimal extended policy N^∗ can be determined analytically or numerically. Finally, a numerical example is given to illustrate some theoretical results of the repair model proposed in this paper.     

A bivariate optimal replacement policy for a cold standby repairable system with preventive repair

In this paper, the repair-replacement problem for a deteriorating cold standby repairable system is investigated. The system consists of two dissimilar components, in which component 1 is the main component with use priority and component 2 is a supplementary component. In order to extend the working time and economize the running cost of the system, preventive repair for component 1 is performed every time interval T, and the preventive repair is “as good as new”. As a supplementary component, component 2 is only used at the time that component 1 is under preventive repair or failure repair. Assumed that the failure repair of component 1 follows geometric process repair while the repair of component 2 is “as good as new”. A bivariate repair-replacement policy (T, N) is adopted for the system, where T is the interval length between preventive repairs, and N is the number of failures of component 1. The aim is to determine an optimal bivariate policy (T, N)^∗ such that the average cost rate of the system is minimized. The explicit expression of the average cost rate is derived and the corresponding optimal bivariate policy can be determined analytically or numerically. Finally, a Gamma distributed example is given to illustrate the theoretical results for the proposed model.     

The Proof of a Conjecture of Additive Number Theory

The aim of this paper is to show that for any n∈N, n>3, there exist a, b∈N* such that n=a+b, the “lengths” of a and b having the same parity (see the text for the definition of the “length” of a natural number). Also we will show that for any n∈N, n>2, n≠5, 10, there exist a, b∈N* such that n=a+b, the “lengths” of a and b having different parities. We will prove also that for any prime p≡7(mod 8) there exist a, b∈N* such that p=a²+b, the “length” of b being an even number.     

Mathematical model of liquid–liquid equilibrium for a ternary system using the GMDH-type neural network and genetic algorithm

A GMDH type-neural network was used to predict liquid phase equilibrium data for the (water + ethanol + trans-decalin) ternary system in the temperature range of 300.2–315.2 K. In order to accomplish modeling, the experimental data were divided into train and test sections. The data set was divided into two parts: 70% were used as data for “training” and 30% were used as a test set. The predicted values were compared with those of experimental values in order to evaluate the performance of the GMDH neural network method. The results obtained by using GMDH type neural network are in excellent agreement with the experimental results.     

On bidding for a fixed number of items in a sequence of auctions

We consider the problem of a firm (“the buyer”) that must acquire a fixed number (L) of items. The buyer can acquire these items either at a fixed buy-it-now price in the open market or by participating in a sequence of N > L auctions. The objective of the buyer is to minimize his expected total cost for acquiring all L items. We model this problem as a Markov Decision Process and establish monotonicity properties for the optimal value function and the optimal bidding strategies.     

Blow-up and global solutions for a class of nonlinear parabolic equations with different kinds of boundary conditions

By constructing different auxiliary functions and using Hopf’s maximum principle, the sufficient conditions for the blow-up and global solutions are presented for nonlinear parabolic equation u_t = ∇(a(u)b(x)c(t)∇u) + f(x, u, q, t) with different kinds of boundary conditions. The upper bounds of the “blow-up time” and the “upper estimates” of global solutions are provided. Finally, some examples are presented as the application of the obtained results.     

Inspection scheduling for imperfect production processes under free repair warranty contract

The paper considers scheduling of inspections for imperfect production processes where the process shift time from an ‘in-control’ state to an ‘out-of-control’ state is assumed to follow an arbitrary probability distribution with an increasing failure (hazard) rate and the products are sold with a free repair warranty (FRW) contract. During each production run, the process is monitored through inspections to assess its state. If at any inspection the process is found in ‘out-of-control’ state, then restoration is performed. The model is formulated under two different inspection policies: (i) no action is taken during a production run unless the system is discovered in an ‘out-of-control’ state by inspection and (ii) preventive repair action is undertaken once the ‘in-control’ state of the process is detected by inspection. The expected sum of pre-sale and post-sale costs per unit item is taken as a criterion of optimality. We propose a computational algorithm to determine the optimal inspection policy numerically, as it is quite hard to derive analytically. To ease the computational difficulties, we further employ an approximate method which determines a suboptimal inspection policy. A comparison between the optimal and suboptimal inspection policies is made and the impact of FRW on the optimal inspection policy is investigated in a numerical example.     

The randomized threshold for the discrete-time Geo/G/1 queue

This paper discusses a discrete-time Geo/G/1 queue, in which the server operates a random threshold policy, namely 〈p, N〉 policy, at the end of each service period. After all the messages are served in the queue exhaustively, the server is immediately deactivated until N messages are accumulated in the queue. If the number of messages in the queue is accumulated to N, the server is activated for services with probability p and deactivated with probability (1 − p). Using the generating functions technique, the system state evolution is analyzed. The generating functions of the system size distributions in various states are obtained. Some system characteristics of interest are derived. The long-run average cost function per unit time is analytically developed to determine the joint optimal values of p and N at a minimum cost.     

On the security analysis of an image scrambling encryption of pixel bit and its improved scheme based on self-correlation encryption

An image scrambling encryption scheme for pixel bits was presented by Ye [Ye GD. Image scrambling encryption algorithm of pixel bit based on chaos map. Pattern Recognit Lett 2010;31:347-54], which can be seen as one kind of typical binary image scrambling encryption considering from the bit-plain of size M × (8N). However, recently, some defects existing in the original image encryption scheme, i.e., Ye’s scheme, have been observed by Li and Lo [Li CQ, Lo KT. Optimal quantitative cryptanalysis of permutation-only multimedia ciphers against plaintext attacks. Signal Process 2011;91:949-54]. In the attack proposed by Li and Lo at least 3 + ⌈log₂(MN)⌉ plain images of size M × N are used to reveal the permutation matrix W = [w(i, k)] (i ∈ {1, 2, … , M}; k ∈ {1, 2, … , 8N}) which can be applied to recover the exact plain image. In the current paper, at first, one type of special plain image/cipher image is used to analyze the security weakness of the original image scrambling scheme under study. The final encryption vectors TM and TN or the decryption vectors TM′ and TN′ are revealed completely according to our attack. To demonstrate the performance of our attack, a quantified comparison is drawn between our attack and the attack proposed by Li and Lo. Compared with Li and Lo’s attack, our attack is more efficient in the general conditions. In particular, when the sizes of images satisfy the condition M = N or M ? 8N, the number of the used plain images/cipher images is at most 9, which is sharply less than 3 + ⌈log₂(MN)⌉ when M and N are of large size. To overcome the weaknesses of the original scheme, in this paper, an improved image scrambling encryption scheme is proposed. In the improved scheme, the idea of the “self-correlation” method is used to resist the chosen-plaintext attack/known-plaintext attack. The corresponding simulations and analyses illustrate that the improved encryption method has good cryptographic properties, and can overcome the weakness of the original image encryption scheme. Finally, farther improvement is briefly presented for the future work.     

Families of relations modelling preferences under incomplete information

Let us consider a preferential information of type preference–indifference–incomparability (P, I, J), with additional information about differences in attractiveness between pairs of alternatives. The present paper offers a theoretical framework for the study of the “level of constraint” of this kind of partial preferential information. It suggests a number of structures as potential models being less demanding than the classical one in which differences in utilities can be used to represent the comparison of differences in attractiveness. The models are characterized in the more general context of families of non-complete preference structures, according to two different perspectives (called “semantico-numerical” and “matrix”). Both perspectives open the door to further practical applications connected with elicitation of the preferences of a decision maker.     

A dynamic adaptive local search algorithm for the circular packing problem

This paper studies the circular packing problem (CPP) which consists of packing n non-identical circles C_i of known radius r_i, i ∈ N = {1, … , n}, into the smallest containing circle C. The objective is to determine the coordinates (x_i, y_i) of the center of C_i, i ∈ N, as well as the radius r and center (x, y) of C. This problem, which is a variant of the two-dimensional open dimension problem, is solved using a two-step, dynamic, adaptive, local search algorithm. At each iteration, the algorithm identifies the set of potential “best local positions” of a circle C_i, i ∈ N, given the positions of the previously packed circles, and determines for each of these positions the coordinates and radius of the smallest containing circle. The “best local position” minimizes the radius of the current containing circle. That is, every time an additional circle is packed, both the center and the radius of the containing circle are dynamically updated, and the smallest containing circle is known. The experimental results reflect the good performance of the algorithm.     

Markov decision processes with exponentially representable discounting

We generalize the geometric discount of finite discounted cost Markov Decision Processes to “exponentially representable”discount functions, prove existence of optimal policies which are stationary from some time N onward, and provide an algorithm for their computation. Outside this class, optimal “N-stationary” policies in general do not exist.     

An economic production lot size model in an imperfect production system

The paper investigates an EPL (Economic Production Lotsize) model in an imperfect production system in which the production facility may shift from an ‘in-control’ state to an ‘out-of-control’ state at any random time. The basic assumption of the classical EPL model is that 100% of produced items are perfect quality. This assumption may not be valid for most of the production environments. More specifically, the paper extends the article of Khouja and Mehrez [Khouja, M., Mehrez, A., 1994. An economic production lot size model with imperfect quality and variable production rate. Journal of the Operational Research Society 45, 1405–1417]. Generally, the manufacturing process is ‘in-control’ state at the starting of the production and produced items are of conforming quality. In long-run process, the process shifts from the ‘in-control’ state to the ‘out-of-control’ state after certain time due to higher production rate and production-run-time.The proposed model is formulated assuming that a certain percent of total product is defective (imperfect), in ‘out-of-control’ state. This percentage also varies with production rate and production-run time. The defective items are restored in original quality by reworked at some costs to maintain the quality of products in a competitive market. The production cost per unit item is convex function of production rate. The total costs in this investment model include manufacturing cost, setup cost, holding cost and reworking cost of imperfect quality products. The associated profit maximization problem is illustrated by numerical examples and also its sensitivity analysis is carried out.     

A new multicriteria approach for the analysis of efficiency in the Spanish olive oil sector by modelling decision maker preferences

The efficiency in production is often analysed as technical efficiency using the production frontier function. Efficiency scores are usually based on distance computations to the frontier in an m + s-dimensional space, where m inputs produce s outputs. In addition, efficiency improvements consider the total consumption of each input. However, in many cases, the “consumption” of each input can be divided into input-consumption sections (ICSs), and trade-off among the ICSs is possible. This share framework can be used for computing efficiency. This analysis provides information about both the total optimal consumption of each input, as does data envelopment analysis, and the most efficient allocation of the “consumption” among the ICSs. This paper studies technical efficiency using this approach and applies it to the olive oil sector in Andalusia (Spain). A non-parametrical methodology is presented, and an input-oriented Multi-Criteria Linear Programming model (MLP) is proposed. The analysis is developed at global, input and ICSs levels, defining the extent of satisfaction achieved at all these levels for each company, in accordance with their own preferences. The companies’ preferences are modelled with their utility function and their set of weights. MLP offers more detailed information to assist decision makers than other models previously proposed in the literature. In addition to this application, it is concluded that there is room for improvement in the olive oil sector, particularly in the management of the skilled labour. Additionally, the solutions with two opposite scenarios indicate that the model is suitable for the intended decision making process.     

Elliptic complex numbers with dual multiplication

Investigated is a number system in which the square of a basis number: (w)², and the square of its additive inverse: (−w)², are not equal. Termed W space, a vector space over the reals, this number system will be introduced by restating defining relations for complex space C, then changing a defining conjugacy relation from conj(z) + z = 0 in the complexes to conj(z) + z = 1 for W space. This change produces a dual-represented vector space consisting of two dual, isomorphic fields, which are unified under one “context-sensitive” multiplication. Fundamental algebraic and geometric properties will be investigated. W space can be interpreted as a generalization of the complexes but is characterized by an interacting duality which seems to produce two of everything: two representations, two multiplications, two norm values, and two solutions to a linear equation. W space will be compared to a previous suggestion of a similar algebra, and then possible applications will be offered, including a W space fractal.     

Considering the isolation set problem

Given a network, G = [N, E] the Isolation Set Problem (ISP) finds the set of arcs, D ⊆ E, that when removed will separate a predefined set of r distinguished nodes [2]. This involves eliminating connections from a specific set of nodes to the rest of a network. In our increasingly interconnected network-centric world, this might be isolating various units from Headquarters; isolating a portion of a computer network to disrupt communications or to quarantine a virus or some other form of cyber attack; or isolating a cell or sub-group in a terrorist or “dark” network, for example.     

Hidden lessons: How a focus on slope-like properties of quadratic functions encouraged unexpected generalizations

This article presents secondary students’ generalizations about the connections between algebraic and graphical representations of quadratic functions, focusing specifically on the roles of the parameters a, b, and c in the general form of a quadratic function, y = ax² + bx + c. Students’ generalizations about these connections led to a surprising finding: two-thirds of the students interviewed identified the parameter a as the “slope” of the parabola. Analysis of qualitative data from interviews and classroom observations led to the development of three focusing phenomena in the classroom environment that inadvertently supported a focus on slope-like properties of quadratic functions: (a) the use of linear analogies, (b) the rise over run method, and (c) viewing a as dynamic rather than static.