Some modified mathematical analytic derivations of the annual total relevant cost of the inventory model with two levels of trade credit in the supply chain system

Several recent studies in supply chain system and related areas explored various economic order quantity (EOQ) models for noninstantaneous deteriorating items with imperfect quality and trade credit financing. In particular, in the year 2007, Teng et al investigated an EOQ model in which the supplier offers the retailer the permissible delay period M and the retailer, in turn, provides the trade credit period N (with ) to his/her customers. The main purpose of this article is twofold: (a) It modifies the annual total relevant cost TVC(T) in the study of Teng et al and presents the correct derivations of TVC(T) by applying mathematical analytic tools and techniques. (b) It exposes some logical and mathematical problems in the proof of Theorem 1 in Teng et al. It also corrects and overcomes all of the errors and shortcomings by systematically presenting the complete and mathematical solution procedures in order to locate all optimal solutions for the model in Teng et al.     

Manufacturer's optimal pricing and lot-sizing policies under trade-credit financing

In the year 2006, Teng et al considered an appropriate economic production quantity (EPQ) model in which the manufacturer receives the supplier's trade credit and provides trade credit to the customer simultaneously. The following two payment methods were discussed by Teng et al: The main purpose of this paper is summarized below: Finally, with a view to further motivating the interested researchers for using the methodology and mathematical analytic techniques in several other contexts in the field, we have chosen to include, in Section 12, a number of related recent works in the field.     

Burst distribution of a linear code

In this note, the distribution of bursts is obtained, as defined by Chien and Teng, for a linear code and its coset.     

GLOBAL ATTRACTIVITY OF A DELAYED RATIO-DEPENDENT COOPERATIVE SYSTEM WITH STAGE STRUCTURE

An autonomous stage-structured ratio-dependent cooperative system,which was proposed by Muhammadhaji,Teng and Abdurahman,is revisited in this paper. By introducing a new lemma and using the iterative method, a set of sufficient conditions which guarantee the global attractivity of the positive equilibrium is obtained. It is shown that the conditions which guarantee the permanence of the system are enough to ensure the global attractivity of the system. Our result not only complements but also supplements one of the main results of Muhammadhaji, Teng and Abdurahman(Permanence and extinction analysis for a delayed ratio-dependent cooperative system with stage structure, Afr. Mat.(2013)DOI 10.1007/sl3370-013-0162-6).     

Some comments on “A simple method to compute economic order quantities”

In this note, we emphasize that the arithmetic–geometric-mean-inequality approach proposed by Teng [Teng, J.T., 2008. A simple method to compute economic order quantities. European Journal of Operational Research. doi:10.1016/j.ejor.2008.05.019] is not a general solution method. Teng’s approach happens to work and give the correct results when the two terms in an objective function are any functions such that their product is a constant. The classical EOQ model works fine since the product of the two terms is indeed a constant! When the product is not a constant, Teng’s approach is of little use. This is exemplified in Comment 1 via solving the EOQ model with complete backorders (where the model is regarded as having two decision variables). Comment 2 is generally valid for an algebraic method when it is used to solve an objective function with two decision variables.     

Permanence and global stability for nonautonomous N-species Lotka-Volterra competitive system with impulses and infinite delays

In this paper, we consider a class of nonautonomous N-species Lotka-Volterra competitive systems with impulses and infinite delays. By developing the methods given in Teng (2002) [25], we give sufficient conditions for permanence and global attractivity of the system.     

PERMANENCE OF ASYMPTOTICALLY PERIODIC MULTISPECIES LOTKA-VOLTERRA COMPETITION PREDATOR-PREY SYSTEM

In this paper, we consider the permanence of asymptotically periodic mul-tispecies Lotka-Volterra competition predator-prey system. By means of the standard comparison theorem, we improve or extend the corresponding results given by Peng and Chen [1], Teng and Li [2], Zhao and Chen [3]. Also, we obtain the conditions which ensure the permanence and global attractivity of asymptotically periodic multispecies competition predator-prey system.     

An inventory model under two levels of trade credit and limited storage space derived without derivatives

This paper tries to incorporate both Huang’s model [Y.F. Huang, Optimal retailer’s ordering policies in the EOQ model under trade credit financing, J. Oper. Res. Soc. 54 (2003) 1011–1015] and Teng’s model [J.T. Teng, On the economic order quantity under conditions of permissible delay in payments, J. Oper. Res. Soc. 53 (2002) 915–918] by considering the retailer’s storage space limited to reflect the real-life situations. That is, we want to investigate the retailer’s inventory policy under two levels of trade credit and limited storage space. Furthermore, we adopt Teng’s viewpoint [J.T. Teng, On the economic order quantity under conditions of permissible delay in payments, J. Oper. Res. Soc. 53 (2002) 915–918] that the retailer’s unit selling price and the purchasing price per unit are not necessarily equal. Then, an algebraic approach is provided and three easy-to-use theorems are developed to efficiently determine the optimal cycle time. Some previously published results of other researchers can be deduced as special cases. Finally, a numerical example is given to illustrate these theorems and managerial insights are drawn.     

A simple method to compute economic order quantities: Some observations

Teng [2] presents an arithmetic–geometric mean method to be applied to determine the optimal lot size for the EOQ/EPQ models, taking into account backorders. Although the arithmetic–geometric mean method is correct, arguments as to when (not) to use the arithmetic–geometric mean inequality as optimization method are not complete. Moreover, this optimization method does not focus on the method for deriving the optimal backorders level. The main purpose of this work is to overcome these shortcomings, presents a discussion of when (not) to use the cost minimization method and derives the optimal backorders level.     

A note on optimal ordering policies when the supplier provides a progressive interest scheme

Goyal et al. [Goyal, S.K., Teng, J.T., Chang, C.T., 2007. Optimal ordering policies when the supplier provides a progressive interest scheme. European Journal of Operational Research 179, 404–413] explore optimal ordering policies when the supplier provides a progressive interest scheme. The main purpose of this paper is fourfold:

(1)

This paper simplifies the total relevant cost per year Z(T) of Goyal et al. (2007) such that we can locate the optimal solutions of Z(T) by an easier way.     

A note on an EOQ model for deteriorating items under supplier credit linked to ordering quantity

This paper explores the inventory replenishment policy for deteriorating items in which the supplier provides a permissible delay to the purchaser if the order quantity is greater than or equal to a predetermined quantity. As a matter of fact, the inventory system discussed by this paper is the same as that of Chang et al. [C.T. Chang, L.Y. Ouyang, J.T. Teng, An EOQ model for deteriorating items under supplier credit credits linked to ordering quantity, Appl. Math. Model. 27 (2003) 983–996]. However, their approach in solving the problems needs further analysis. This article deals with an alternative approach to present a simple procedure in order to determine the optimal ordering policy when the supplier provides a permissible delay in payments linked to order quantity. Numerical examples reveal that the solution algorithm described in this paper is accurate and rapid.     

Global dynamics of a class of SEIRS epidemic models in a periodic environment

In this paper, we study a class of periodic SEIRS epidemic models and it is shown that the global dynamics is determined by the basic reproduction number R₀ which is defined through the spectral radius of a linear integral operator. If R₀<1, then the disease free periodic solution is globally asymptotically stable and if R₀>1, then the disease persists. Our results really improve the results in [T. Zhang, Z. Teng, On a nonautonomous SEIRS model in epidemiology Bull. Math. Biol. 69 (8) (2007) 2537-2559] for the periodic case. Moreover, from our results, we see that the eradication policy on the basis of the basic reproduction number of the time-averaged system may overestimate the infectious risk of the periodic disease. Numerical simulations which support our theoretical analysis are also given.     

Optimal Ordering Policy for Deteriorating Items with Partial Backlogging under Permissible Delay in Payments

In 1985, Goyal developed an Economic order quantity (EOQ) model under conditions of permissible delay in payments. Jamal et al. then generalized Goyal’s model for deteriorating items with completely backlogging. However, they only ran several simulations to indicate that the total relevant cost may be convex. Recently, Teng amended Goyal’s model by considering the difference between unit price and unit cost, and provided an alternative conclusion that it makes economic sense for some retailers to order less quantity and take the benefits of the permissible delay more frequently. However, he did not consider deteriorating items and partial backlogging. In this paper, we establish a general EOQ model for deteriorating items when the supplier offers a permissible delay in payments. For generality, our model allows not only the partial backlogging rate to be related to the waiting time but also the unit selling price to be larger than the unit purchase cost. Consequently, the proposed model includes numerous previous models as special cases. In addition, we mathematically prove that the total relevant cost is strictly pseudo-convex so that the optimal solution exists and is unique. Finally, our computational results reveal six managerial phenomena.     

Anchorage strength models for end-debonding predictions in RC beams strengthened with FRP composites

The increase in the flexural capacity of RC beams obtained by externally bonding FRP composites to their tension side is often limited by the premature and brittle debonding of the external reinforcement. An in-depth understanding of this complex failure mechanism, however, has not yet been achieved. With specific regard to end-debonding failure modes, extensive experimental observations reported in the literature highlight the important distinction, often neglected in strength models proposed by researchers, between the peel-off and rip-off end-debonding types of failure. The peel-off failure is generally characterized by a failure plane located within the first few millimetres of the concrete cover, whilst the rip-off failure penetrates deeper into the concrete cover and propagates along the tensile steel reinforcement. A new rip-off strength model is described in this paper. The model proposed is based on the Chen and Teng peel-off model and relies upon additional theoretical considerations. The influence of the amount of the internal tensile steel reinforcement and the effective anchorage length of FRP are considered and discussed. The validity of the new model is analyzed further through comparisons with test results, findings of a numerical investigation, and a parametric study. The new rip-off strength model is assessed against a database comprising results from 62 beams tested by various researchers and is shown to yield less conservative results. Russian translation published in Mekhanika Kompozitnykh Materialov, Vol. 44, No. 3, pp. 373–388, May–June, 2008.     

Partial backlogging inventory lot-size models for deteriorating items with fluctuating demand under inflation

In this paper, the traditional inventory lot-size model is extended to allow not only for general partial backlogging rate but also for inflation. The assumptions of equal cycle length and constant shortage length imposed in the model developed by Moon et al. [Moon, I., Giri, B.C., Ko, B., 2005. Economic order quantity models for ameliorating/deteriorating items under inflation and time discounting, European Journal of Operational Research 162(3), 773–785] are also relaxed. For any given number of replenishment cycles the existence of a unique optimal replenishment schedule is proved and further the convexity of the total cost function of the inventory system in the number of replenishments is established. The theoretical results here amend those in Yang et al. [Yang, H.L., Teng, J.T., Chern, M.S., 2001. Deterministic inventory lot-size models under inflation with shortages and deterioration for fluctuating demand, Naval Research Logistics 48(2), 144–158] and provide the solution to those two counterexamples by Skouri and Papachristos [Skouri, K., Papachristos, S., 2002. Note on “deterministic inventory lot-size models under inflation with shortages and deterioration for fluctuating demand” by Yang et al. Naval Research Logistics 49(5), 527–529.]. Finally we propose an algorithm to find the solution, and obtain some managerial results by using sensitivity analyses.     

Inductivism and Parmenidean epistemology: Kyburg’s way

cording to Henry Kyburg, all extralogical and extramathematical propositions accepted as evidence and all propositions accepted inductively on the basis of such evidence are uncertain. There is a possibility of error. Consequently, neither the corpus of inductively accepted statements nor the corpus of statements accepted as evidence can serve as a standard for serious possibility in the sense I have deployed since the 1970s. The standard for serious possibility remains an unchanging Parmenidean standard. In contrast to other Parmenidean epistemologists that eschew inductive acceptances Kyburg insists that the corpus of evidence and of inductively accepted statements is subject to critical review and change; but the changes have no bearing on the standard for serious possibility.have always agreed with Henry’s emphasis on a distinction between acceptance as evidence and inductive acceptance. But I have insisted that the corpus of evidence or state of full belief is a standard for serious possibility and that the standard is subject to modification.burg does think of acceptance as evidence and inductive acceptance as modal notions and has recently used the expression “serious possibility” in this connection. But when Kyburg and Teng speak of “risky knowledge”, they are speaking of claims that might be false in the sense of serious possibility that they seem to be suggesting is immune to change and seems to correlate with serious possibility as I have used it since the 1970s. So acceptance (both inductive and evidential) are modal notions subject to change but are not to be confused with the notion of serious possibility of error or riskiness.     

Smoothed analysis of termination of linear programming algorithms

 We perform a smoothed analysis of a termination phase for linear programming algorithms. By combining this analysis with the smoothed analysis of Renegar's condition number by Dunagan, Spielman and Teng (http://arxiv.org/abs/cs.DS/0302011) we show that the smoothed complexity of interior-point algorithms for linear programming is O(m ³ log(m/Σ)). In contrast, the best known bound on the worst-case complexity of linear programming is O(m ³ L), where L could be as large as m. We include an introduction to smoothed analysis and a tutorial on proof techniques that have been useful in smoothed analyses. Received: December 10, 2002 / Accepted: April 28, 2003 Published online: June 5, 2003 Key words. smoothed analysis – linear programming – interior-point algorithms – condition numbers Mathematics Subject Classification (1991): 90C05, 90C51, 68Q25     

Combinatorial aspects of geometric graphs

As a special case of our main result, we show that for all L> 0, each k-nearest neighbor graph in d dimensions excludes K_h as a depth L minor if h = Ω(L^d). More generally, we prove that the overlap graphs defined by Miller, Teng, Thurston and Vavasis (1993) have this combinatorial property. By a construction of Plotkin, Rao and Smith (1994), our result implies that overlap graphs have “good” cut-covers, answering an open question of Kaklamanis, Krizanc and Rao (1993). Consequently, overlap graphs can be emulated on hypercube graphs with a constant factor of slow-down and on butterfly graphs with a factor of O(log^* n) slow-down. Therefore, computations on overlap graphs, such as finite element and finite difference methods on “well-conditioned” meshes and image processing on k-nearest neighbor graphs, can be performed on hypercubic parallel machines with a linear speed-up. Our result, in conjunction with a result of Plotkin, Rao and Smith, also yields a combinatorial proof that overlap graphs have separators of sublinear size. We also show that with high probability, the Delaunay diagram, the relative neighborhood graph, and the k-nearest neighbor graph of a random point set exclude K_h as a depth L minor if h = Ω(L^d/2 log n).     

The persistence of a general nonautonomous single-species Kolmogorov system with delays

In this paper, we establish some sufficient conditions for the persistence of system x^′(t)=x(t)f(t,_xt)$x^{\prime }\left(t\right)=x\left(t\right)f(t,x_t)$ without the condition that f(t,φ)$f(t,\phi )$ is monotonously decreasing with respect to φ$\phi$. This partly answers the open problems proposed by Teng.     

Further investigations into the stability and bifurcation of a discrete predator–prey model

In this paper we revisit a discrete predator–prey model with a non-monotonic functional response, originally presented in Hu, Teng, and Zhang (2011) [6]. First, by citing several examples to illustrate the limitations and errors of the local stability of the equilibrium points _E3$E_3$ and _E4$E_4$ obtained in this article, we formulate an easily verified and complete discrimination criterion for the local stability of the two equilibria. Here, we present a very useful lemma, which is a corrected version of a known result, and a key tool in studying the local stability and bifurcation of an equilibrium point in a given system. We then study the stability and bifurcation for the equilibrium point _E1$E_1$ of this system, which has not been considered in any known literature. Unlike known results that present a large number of mathematical formulae that are not easily verified, we formulate easily verified sufficient conditions for flip bifurcation and fold bifurcation, which are explicitly expressed by the coefficient of the system. The center manifold theory and Project Method are the main tools in the analysis of bifurcations. The theoretical results obtained are further illustrated by numerical simulations.