Kit-Nam Francis Leung

A generalized geometric-programming solution to “An economic production quantity model with flexibility and reliability considerations”

The classical economic production quantity (EPQ) model assumes that items are produced by a perfectly reliable production process with a fixed set-up cost. While the reliability of the production process cannot be perfected cost-free, the set-up cost can be reduced by investment in flexibility improvement. In this paper, we propose an EPQ model with a flexible and imperfect production process. We formulate this inventory decision problem using geometric programming (GP), establish more general results using the arithmetic-geometric mean inequality, and solve the problem to obtain a closed-form optimal solution. Following the theoretical treatment, we provide a numerical example to demonstrate that GP has potential as a valuable analytical tool for studying a certain class of inventory control problems. Finally we discuss some aspects of sensitivity analysis of the optimal solution based on the GP approach.

Analysis for a two-dissimilar-component cold standby repairable system with repair priority

Abstract In this paper, a cold standby repairable system consisting of two dissimilar components and one repairman is studied. Assume that working time distributions and repair time distributions of the two components are both exponential, and Component 1 has repair priority when both components are broken down. After repair, Component 1 follows a geometric process repair while Component 2 obeys a perfect repair. Under these assumptions, using the perfect repair model, the geometric process repair model and the supplementary variable technique, we not only study some important reliability indices, but also consider a replacement policy T, under which the system is replaced when the working age of Component 1 reaches T. Our problem is to determine an optimal policy T⁎ such that the long-run average loss per unit time (i.e. average loss rate) of the system is minimized. The explicit expression for the average loss rate of the system is derived, and the corresponding optimal replacement policy T⁎ can be found numerically. Finally, a numerical example for replacement policy T is given to illustrate some theoretical results and the models applicability.

Using the complete squares method to analyze a lot size model when the quantity backordered and the quantity received are both uncertain

Several researchers have recently derived formulae for economic order quantities (EOQs) with some variants without reference to the use of derivatives, neither for first-order necessary conditions nor for second-order sufficient conditions. In addition, this algebraic derivation immediately produces an individual formula for evaluating the minimum expected annual cost. The purpose of this paper is threefold. First, this study extends the previous result to the EOQ formula, taking into account the scenario where the quantity backordered and the quantity received are both uncertain. Second, the complete squares method can readily derive global optimal expressions from a non-convex objective function in an algebraic manner. Third, the explicit identification of some analytic cases can be obtained: it is not as easy to do this using decomposition by projection. A numerical example has been solved to illustrate the solution procedure. Finally, some special cases can be deduced from the EOQ model under study, and concluding remarks are drawn.

A generalization of sensitivity of the inventory model with partial backorders

In this paper, we use the elementary techniques of differential calculus to investigate the sensitivity analysis of Montgomery et al.s [Montgomery, D.C., Bazaraa, M.S., Keswani, A.K., 1973. Inventory models with a mixture of backorders and lost sales. Naval Research Logistics Quarterly 20, 225-263] inventory model with a mixture of backorders and lost sales and generalize Chu and Chungs [Chu, P., Chung, K.J., 2004. The sensitivity of the inventory model with partial backorders. European Journal of Operational Research 152, 289-295] sensitivity analysis. We provide three numerical examples to demonstrate our findings, and remark the interpretation of the global minimum of the average annual cost at which the complete backordering occurs.

A sequential method for preventive maintenance and replacement of a repairable single-unit system

This paper is concerned with when to implement preventive maintenance (PM) and replacement for a repairable ‘single-unit’ system in use. Under the main assumption that a ‘single-unit’ system gradually deteriorates with time, a sequential method is proposed to determine an optimal PM and replacement strategy for the system based on minimising expected loss rate. According to this method, PM epochs are determined one after the other, and consequently we can make use of all previous information on the operation process of the system. Also the replacement epoch depends on the effective age of the system. A numerical example shows that the sequential method can be used to solve the PM and replacement problem of a ‘single-unit’ system efficiently. Some properties of the loss functions W(L̄n,b̄n) and W̄r(N) with respect to PM and replacement respectively are discussed in the appendix.

A note on ‘A bivariate optimal replacement policy for a repairable system’

An error appearing in equation (3) of Y.L. Zhang (J. Appl. Prob., 1994, 31, 1123–1127) has been pointed out by S.H. Sheu (Eur. J. Oper. Res., 1999, 112, 503–516) and the correct expressions (25)–(27) given accordingly on pp. 510–511. However, the derivation of the key expression (27), the long-run expected loss rate, was not presented. The purpose of this note is threefold. First, since a monotone process (e.g. an arithmetic, geometric, or arithmetic–geometric process) approach, as discussed by K.N.F. Leung (Eng. Optimiz., 2001, 33, 473–484), is considered to be relevant, realistic, and appropriate to the modelling of a deteriorating system maintenance problem, it is worth explicitly developing this expression, which is of benefit to the subsequent studies. Secondly, equation (3) in Zhang (1994) is shown to be fundamentally correct and so it can be viewed as an alternative method of formulating similar bivariate cases. Thirdly, although equations (4) and (5) in Zhang (1994) have been logically and correctly derived, both can be readily reduced to their simplest forms which are derived here.

Optimal production lot sizing with backlogging, random defective rate, and rework derived without derivatives

Abstract Several researchers have recently derived formulae for economic production quantities (EPQs) with some variants without reference to the use of derivatives, neither for first-order necessary conditions nor for second-order sufficient conditions. In addition, this algebraic derivation immediately produces an individual formula for evaluating the minimum expected annual cost. The purpose of this note is twofold. First, this study extends earlier results to the EPQ formula, taking the imperfect rework process into account. Second, the algebraic complete-squares and perfect-squares methods can readily derive optimal expressions from an objective function in a more simple, direct, and natural manner than the algebraic method of unity decomposition adopted earlier. A numerical example has been solved to illustrate the solution procedure, and some remarks are made to conclude the note.

An integrated production-inventory system with lot streaming and complete backordering in a three-stage multi-firm supply chain solved algebraically

We formulate an integrated model of a three-stage multi-firm supply chain based on an integer multiplier at each stage, lot streaming allowed for all suppliers and manufacturers, and complete backordering allowed for some/all retailers. Then we derive the optimal solution to the integrated model using the methods of complete squares and perfect squares. These are simple algebraic approaches so that ordinary readers unfamiliar with differential calculus can understand the optimal solution procedure with ease. For our model, we also need check that the optimal solution, which is algebraically derived, is a global one. We solve a numerical example to illustrate the procedure. We finally deduce Ben-Daya and Al-Nassar’s (2008) models and remark on extending to a higher stage multi-firm supply chain from the integrated model.

Two general forms of approximation for determining near optimal inspection intervals with non-zero inspection and replacement times in a deteriorating production system

The construction and solution of the next simplest but more realistic profit model for an exponential shift (or failure) distribution were presented in Ben-Daya and Hariga (B&H) [Ben-Daya, M., & Hariga, M. (1998). A maintenance inspection model: Optimal and heuristic solutions. International Journal of Quality and Reliability Management, 15, 481-488], in which the model presented looks much more complicated than the simplest profit model proposed by Baker [Baker, M. J. C. (1990). How often should a machine be inspected? International Journal of Quality and Reliability Management, 7, 14-18]. However, once we confer economic meanings to some mathematical expressions and rewrite the whole model, the revised model not only becomes less complicated and more comprehensible than the original one but also draws a closer analogy with Bakers model and the associated solution algorithms. We also develop two distinct sets of general explicit formulae and present two corresponding heuristic algorithms for solving B&Hs model. The examples show that either of these new algorithms for determining a near optimal inspection interval for a deteriorating (or an imperfect) production system is not only more close to the optimum but also more efficient computationally than the one suggested by B&H (1998), and can give the value to any required degree of accuracy. The main purpose of this article is threefold: (1) to confer economic meanings to some expressions, (2) to rewrite the model and the Proof of Theorem 1, and (3) to introduce general exponential and logarithmic forms of approximation for further approximating optimal inspection intervals and to give merits and demerits of the two algorithms.

Optimal replacement policy for a two-component cold standby system with repair priority

Abstract In this paper, a cold standby repairable system consisting of two dissimilar components and one repairman is examined. It was assumed that both Component 1 and Component 2, were not as good as new after repair, and Component 1 had repair priority. Under these assumptions, using the geometric process repair model, a discrete replacement policy N under which the system is replaced when the number of failures of Component 1 reaches N is considered. The problem was to determine an optimal policy N∗ so that the long-run average loss per unit time L(N) of the system was minimized. The expression L(N) was explicitly derived, and the corresponding optimal policy N∗ can be determined by minimizing L(N) with respect to N. A numerical example and the associated MATLAB computer program are given to illustrate the model—s applicability. Finally, conclusions are drawn for this study.