An integrated production-inventory system in a multi-stage multi-firm supply chain

We first generalize a number of integrated models with/without lot streaming and with/without complete backorders under the integer-multiplier coordination mechanism, and then individually derive the optimal solution to the three- and four-stage model, using algebraic methods of complete squares and perfect squares. We subsequently deduce optimal expressions for some well-known models. For our model, we check that the optimal solution, which is algebraically derived, is a global one. We present three numerical examples for illustrative purposes. We finally suggest some future research work involving extension or modification of the generalized model.Production-inventory Lot streaming Complete backorders The complete/perfect squares method

A supplement to "A generalized algebraic model for optimizing inventory decisions in a centralized or decentralized multi-stage multi-firm supply chain"

In this supplement, a number of integrated models under the integer multiplier coordination mechanism is generalized by allowing complete backorders penalized by not only linear but also fixed costs for some/all retailers. The optimal solution to such a three-stage generalized model is algebraically derived, from which optimal expressions for some well-known models are deduced. In addition, an expression for sharing the coordination benefits is modified, and a numerical example for illustrative purposes is presented. A ready future research work involving extension of the generalized model is suggested to conclude the supplement.Production-inventory Lot streaming Inspection Complete backorders The complete/perfect squares method

A technical note on "Optimizing inventory decisions in a multi-stage multi-customer supply chain"

We first generalize Khouja [Khouja, M., 2003. Optimizing inventory decisions in a multi-stage multi-customer supply chain. Transportation Research Part E: Logistics and Transportation Review 39 (3), 193-208] integrated model considering the integer multipliers mechanism and next individually derive the optimal solution to the three- and four-stage model using the perfect squares method, which is a simple algebraic approach so that ordinary readers unfamiliar with differential calculus can understand the optimal solution procedure with ease. We subsequently deduce the optimal expressions for Khouja (2003) and Cárdenas-Barrón [Cárdenas-Barrón, L.E., 2007. Optimal inventory decisions in a multi-stage multi-customer supply chain: a note. Transportation Research Part E: Logistics and Transportation Review 43 (5), 647-654] model, and identify the associated errors in Khouja (2003). We present two numerical examples for illustrative purposes. We finally shed light on some future research by extending or modifying the generalized model.Inventory Production Without derivatives The perfect squares method

A generalized algebraic model for optimizing inventory decisions in a centralized or decentralized multi-stage multi-firm supply chain

First of all, a number of integrated models with/without lot streaming under the integer multiplier coordination mechanism is generalized by allowing lot streaming and three types of inspection for some/all upstream firms. Secondly, the optimal solutions to the three- and four-stage models are individually derived, both using the perfect squares method, which is a simple algebraic approach so that ordinary readers unfamiliar with differential calculus can easily understand how to obtain the optimal solution procedures. Thirdly, optimal expressions for some well-known models are deduced. Fourthly, expressions for sharing the coordination benefits based on Goyal's (1976) scheme are derived, and a further sharing scheme is introduced. Fifthly, two numerical examples for illustrative purposes are presented. Finally, some future research works involving extension or modification of the generalized model are suggested.Production-inventory Lot streaming Inspection The perfect squares method

Technical note: A use of the complete squares method to solve and analyze a quadratic objective function with two decision variables exemplified via a deterministic inventory model with a mixture of backorders and lost sales

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 average annual cost. The purpose of this paper is twofold. Exemplifying a use of the complete squares method through solving and analyzing Montgomery et al.'s [Montgomery, D.C., Bazaraa, M.S., Keswani, A.C., 1973. Inventory models with a mixture of backorders and lost sales. Naval Research Logistics Quarterly 20, 255-263] model, i.e. the EOQ model taking into account the case of partial backordering first we can readily derive global optimal expressions from a non-convex quadratic cost function with two decision variables in an algebraic manner, second we can straightforwardly identify some analytic cases in a way that is not as easy to do this using calculus. 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.

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.Inventory Production Complete backorders Without derivatives The complete squares method The global solution

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 Chung's [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.Inventory model Partial backorders Sensitivity analysis Monotonic properties