A simple method to compute economic order quantities

In this note, a simple method by using the arithmetic–geometric-mean-inequality theorem is proposed to computer the global minimum economic order quantities without taking complex differential calculus or using tedious algebraic manipulations. In contrast to (Minner, S., 2007. A note on how to compute economic order quantity without derivatives by cost comparisons. International Journal of Production Economics 105, 293–296; Wee, H.M., Wang, W.T., Chung, C.J., 2009. A modified method to computer economic order quantities without derivatives by cost-difference comparisons. European Journal of Operational Research) based on a local cost minimum initially to derive the solution and then proven it’s the global minimum, the proposed method yields the global minimum cost immediately and explicitly without using the cost comparisons and letting the time horizon to infinity.     

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.     

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.     

Finding economic order quantities with nonlinear goal programming

This paper applies nonlinear goal programming (NLGP) to obtain the economic order quantity (EOQ) in a multi-item inventory problem. The model demonstrates how the appropriate priority structure can be selected to determine the optimum EOQ. Sensitivity analysis on the weight structure in a priority structure of the goals has been performed to obtain different solutions in the decision-making environment. The ideal solution is the identified among the solutions associated with different weight structures. The minimum ofd ₁ distances from different solutions to the ideal solution identifies the most acceptable solution. The associated weight structure will be the appropriate weight structure according to the decision-making situation. The effectiveness of the NLGP model is demonstrated via an example.     

An economic order quantity model for deteriorating items with partially permissible delay in payments linked to order quantity

To attract more sales suppliers frequently offer a permissible delay in payments if the retailer orders more than or equal to a predetermined quantity W. In this paper, we generalize [Goyal, S.K., 1985. EOQ under conditions of permissible delay in payments. Journal of the Operational Research Society 36, 335–338] economic order quantity (EOQ) model with permissible delay in payment to reflect the following real-world situations: (1) the retailer’s selling price per unit is significantly higher than unit purchase price, (2) the interest rate charged by a bank is not necessarily higher than the retailer’s investment return rate, (3) many items such as fruits and vegetables deteriorate continuously, and (4) the supplier may offer a partial permissible delay in payments even if the order quantity is less than W. We then establish the proper mathematical model, and derive several theoretical results to determine the optimal solution under various situations and use two approaches to solve this complex inventory problem. Finally, a numerical example is given to illustrate the theoretical results.     

A note on the economic lot size of the integrated vendor–buyer inventory system derived without derivatives

Numerous researches on the integrated production inventory models use differential calculus to solve the multi-variable problems. This study simplifies the solution procedure using a simple algebraic method to solve the multi-variable problems. As a result, students who are unfamiliar with calculus may be able to understand the solution procedure with ease. This paper refers to the approach by Grubbström and Erdem [R.W. Grubbström, A. Erdem, The EOQ with backlogging derived without derivatives, International Journal of Production Economics 59 (1999) 529–530] and extends the model by Yang and Wee [P.C. Yang, H.M. Wee, The economic lot size of the integrated vendor–buyer system derived without derivatives. Optimal Control Applications and Methods 23 (2002) 163–169] to derive an algebraic method to solve the three decision variables of the proposed model.     

An entropic economic order quantity (EnEOQ) for items with imperfect quality

This paper presents an entropic version of an EOQ model with imperfect quality items. The approach adopted herein models the commodity flow (demand rate) as a heat flow in a thermodynamic system. As a result, an entropy cost term is added to the classical inventory cost to form an entropic total inventory cost function. This provides an estimation of the hidden or difficult to estimate cost inventory systems that usually are the result of disorder (or entropy). A mathematical model is developed with numerical results presented and discussed.     

The economic lot size of the integrated vendor-buyer inventory system derived without derivatives: A simple derivation

In this paper, we first complement an inappropriate mathematical error on the total cost in two previous published research papers in inventory field. Then an arithmetic-geometric inequality method is proposed not only to simplify the algebraic method of completing prefect squares, but also to complement their shortcomings. In addition, we also provide a closed-form optimal solution to the integrated vendor-buyer inventory system without using complex derivatives. Finally, the proposed method seems to be easy-to-understand and simple-to-apply by individuals who are not familiar with differential calculus.     

A new solution method for the finite-horizon discrete-time EOQ problem

We consider the finite-horizon discrete-time economic order quantity problem. Kovalev and Ng (2008) have developed a solution approach for solving this problem. Their approach requires a search for the optimal number of orders, which takes O(logn)$\text{O}(\log n)$ time. In this note, we present a modified solution method, which can determine the optimal solution without the need of such a search.     

Determining order quantities in multi-product inventory systems subject to multiple constraints and incremental discounts

This paper presents a mixed-integer programming model for ordering items in multi-product multi-constraint inventory systems from suppliers who offer incremental quantity discounts. The model is based on a piecewise linear approximation of the number of orders function. It allows any number of linear constraints and determines if independent or common (fixed) cycle ordering has a lower total cost. The paper discusses implementation issues and presents results of computational tests on example problems from the operational research literature as well as randomly generated test problems.     

Optimizing the economic lot size of a three-stage supply chain with backordering derived without derivatives

Multi-stage supply chain management integration provides a key to successful international business operations. This is because the integrated approach improves the global system performance and cost efficiency. The integrated production inventory models using differential calculus to solve the multi-variable problems are prevalent in operational research. This paper extends the integrated vendor–buyer inventory problem by Yang and Wee [Yang, P.C., Wee, H.M., 2002. The economic lot size of the integrated vendor–buyer system derived without derivatives. Optimal Control Applications and Methods 23, 163–169] to solve the multi-variable problems in the supply chain, and simplifies the solution procedure using a simple algebraic method. As a result, the solution procedure may be easily understood and applied so as to derive the optimal solution.     

A discontinuous Galerkin finite element method for time dependent partial differential equations with higher order derivatives

In this paper, we develop a new discontinuous Galerkin (DG) finite element method for solving time dependent partial differential equations (PDEs) with higher order spatial derivatives. Unlike the traditional local discontinuous Galerkin (LDG) method, the method in this paper can be applied without introducing any auxiliary variables or rewriting the original equation into a larger system. Stability is ensured by a careful choice of interface numerical fluxes. The method can be designed for quite general nonlinear PDEs and we prove stability and give error estimates for a few representative classes of PDEs up to fifth order. Numerical examples show that our scheme attains the optimal -th order of accuracy when using piecewise -th degree polynomials, under the condition that is greater than or equal to the order of the equation.

     

A comprehensive note on: An economic order quantity with imperfect quality and quantity discounts

Lin [T.Y. Lin, An economic order quantity with imperfect quality and quantity discounts, Appl. Math. Model. 34 (10) (2010) 3158–3165] recently proposed an EOQ model with imperfect quality and quantity discounts, where the lot-splitting shipments policy is adopted. In this note we first rectify the holding cost terms showed in Lin to obtain a new objective function, then resolve the problem and develop an easy to implement algorithm to find the overall optimal solutions for the model. Besides, we present a new model for items with imperfect quality, where lot-splitting shipments and different holding costs for good and defective items are considered. The closed-form formulas for determining the optimal ordering and shipping policies are derived. Also, the results are examined analytically and numerically to gain more insights of the solutions.     

A novel model of nonlocal thermoelasticity with time derivatives of higher order

The objective of this work is to introduce a new system of differential equations describing the nonlocal thermoelasticity theory with higher time derivatives and two-phase lags. In order to obtain this model, we used the nonlocal continuum theory proposed by Eringen and the methodology of the Taylor series expansion of higher-order time derivatives. Some generalized thermoelasticity theories follow as limited cases. This model is used to study the thermoelastic interaction in a nonlocal medium. The medium is exposed to an applied magnetic field and a periodic time heat source with a constant strength. Some comparisons have been displayed in figures to estimate the influences of the nonlocal parameter and magnetic field as well as the parameters of higher-order on all the field quantities.     

The EOQ model with defective items and partially permissible delay in payments linked to order quantity derived analytically in the supply chain management

A lot of researchers develop their inventory models under trade credit by assuming that the supplier offers the retailer fully permissible delay in payments and the products received are all non-defective. However, from the viewpoint of practice, it can often be found that the supplier offers the retailer a fully permissible delay in payments only when the order quantity is greater than or equal to the specific quantity. Furthermore, the products received usually contain some defective items. This paper establishes the EOQ model with defective items and partially permissible delay in payments linked to order quantity. It also uses the rigorous method of mathematics to derive the solution procedure to locate the optimal solution. Finally, numerical examples are given to illustrate all theoretical results in this paper.     

A characterization of parallelepipeds related to weak derivatives

Summary We characterize parallelepipeds in R^m within the family of all convex bodies by a property of special measures on its boundary. We show that these measures are related to weak derivatives (in the sense of [5] and [8]) of convex-valued functions. The results can be applied (see [9]) to derive a generalization of a theorem of Lehmann (see [4]) on the comparison of uniform location experiments.     

A numerical method to price European derivatives based on the one factor LIBOR Market Model of interest rates

We consider the problem of pricing European interest rate derivatives based on the LIBOR Market Model (LMM) with one driving factor. We derive a closed-form approximation of the transition probability density functions associated to the stochastic dynamical systems that describe the behaviour of the forward LIBOR interest rates in the LMM. These approximate formulae are based on a truncated power series expansion of the solutions of the Fokker–Planck equations associated to the LMM. The approximate probability density functions obtained are used to price European interest rate derivatives using the method of discounted expectations. The resulting integrals are low dimensional when the most commonly traded European interest rate derivatives are considered, and they can be computed efficiently using elementary numerical quadrature schemes (i.e. Simpson’s rule). The algorithm obtained is very well suited for parallel computing and is tested on the problem of pricing several derivatives including an European swaption and an interest rate spread option. In both cases, the method proposed in this paper appears to be accurate (i.e. relative error of order 10⁻², 10⁻³, or even 10⁻⁴) and approximately between 278 and 63 000 times faster than previous methods based on the Monte Carlo simulation of the LMM stochastic dynamical systems.

The website http://www.econ.univpm.it/pacelli/ballestra/finance/w2 contains material that helps the understanding of this paper and makes available to the interested users the computer programs that implement the numerical method proposed.