Note on sensitivity analysis of inventory model with partial backorders

In reality, most of the values used for estimating future inventory are parameters derived from the present inventory model. Therefore, researchers must consider how changes in related environments will impact these parameters. A sensitivity analysis explores the impact on the optimal solution when the parameter values vary. This note extends Chu and Chung’s [European Journal of Operational Research 152 (2004) 289–295] paper for the sensitivity analysis of the inventory model with partial backorders. We found the criterion for determining when to run the inventory system. This new discovery deserves careful examination. Numerical examples are provided to demonstrate our findings.     

A deteriorating inventory model with time-varying demand and shortage-dependent partial backlogging

Recently, Chu et al. [P. Chu, K.L. Yang, S.K. Liang, T. Niu, Note on inventory model with a mixture of back orders and lost sales, European Journal of Operational Research 159 (2004) 470–475] presented the necessary condition of the existence and uniqueness of the optimal solution of Padmanabhan and Vrat [G. Padmanabhan, P. Vrat, Inventory model with a mixture of back orders and lost sales, International Journal of Systems Science 21 (1990) 1721–1726]. However, they included neither the purchase cost nor the cost of lost sales into the total cost. In this paper, we complement the shortcoming of their model by adding not only the cost of lost sales but also the non-constant purchase cost, and then extend their model from a constant demand function to any log-concave demand function. We also provide a simple solution procedure to find the optimal replenishment schedule. Further, we use a couple of numerical examples to illustrate the results and conclude with suggestions for future research.     

A note on “Two-warehouse inventory model with deterioration under FIFO dispatch policy”

A recently published paper by Lee [C.C. Lee, Two-warehouse inventory model with deterioration under FIFO dispatching policy, European Journal of Operational Research 174 (2006) 861–873] considers different dispatching models for the two-warehouse inventory system with deteriorating items, in which Pakkala and Achary’s LIFO (last-in–first-out) model [T.P.M. Pakkala, K.K. Achary, A deterministic inventory model for deteriorating items with two warehouses and finite replenishment rate, European Journal of Operational Research 57 (1992) 71–76] is first modified, and then the author concludes that the modified LIFO model always has a lower cost than Pakkala and Achary’s LIFO model under a particular condition specified by him. The present note points out that this conclusion is incorrect and misleading. Alternatively, we provide a new sufficient condition such that the modified LIFO model always has a lower cost than Pakkala and Achary’s model. Besides, we also compare Pakkala and Achary’s original LIFO model with Lee’s FIFO (first-in–first-out) model for the special case where the two warehouses have the same deteriorating rates. Finally, numerical examples are provided to investigate and examine the impact of corresponding parameters on policy choice. The results in this note give a much clearer picture than those at Lee’s paper about the relationships between the different dispatching policies for the two-warehouse inventory system with deterioration items.     

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.     

Research note on the criteria for the optimal solution of the inventory model with a mixture of partial backordering and lost sales

Perishable products are commonly seen in inventory management. By allowing shortages and backlogging, the impact on the cost from the decay of the products can be balanced out. In a recent paper published in Computers and Industrial Engineering [P.L. Abad, Optimal lot size for a perishable good under conditions of finite production and partial backordering and lost sale, Comput. Ind. Eng. 38 (2000) 457–465] considered a problem in such context. However, his algorithm was incomplete due to flaws in his solution procedure. The purpose of this note is to explore the same production inventory models with a mixture of partial backordering and lost sales for deteriorated items. We find the criteria for the optimal solution for different cases and derive a formulated minimum value. By theoretical analysis, we develop a few lemmas to reveal parameter effects and optimal solution procedure. The solutions are illustrated by solving the same examples from Abad’s paper to illustrate the accuracy and completeness of our procedure.     

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.     

An inventory model where backordered demand ratio is exponentially decreasing with the waiting time

We analyze an inventory system with a mixture of backorders and lost sales, where the backordered demand rate is an exponential function of time the customers wait before receiving the item. Stockout costs (backorder cost and lost sales cost) include a fixed cost and a cost proportional to the length of the shortage period. A procedure for determining the optimal policy and the maximum inventory profit is presented. This work extends several inventory models of the existing literature.     

On the inventory model with continuous and discrete lead time,backorders and lost sales

A well-established global supply chain system plays an important role in an enterprise’s strategic management, and an effective just-in-time system is a critical success factor for a global supply chain system. The advantages and benefits of just-in-time are associated with controlling the lead time. In this paper, we propose an approach to overcome the drawback of traditional methods for improving the continuous and discrete lead time with mixture of backorders and lost sales, by which only a local optimal solution can be obtained. In addition, the proposed model allows decision-makers to add suitable constraints to their model in accordance with their actual business environments. Finally, in order to demonstrate the accuracy and effectiveness of the suggested model, a real-life case, some numerical examples and sensitivity analysis are also included.     

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.     

A note on multi-criteria ABC inventory classification using weighted linear optimization

Ramanathan [R. Ramanathan, ABC inventory classification with multiple-criteria using weighted linear optimization, Computers & Operations Research 33 (2006) 695–700] recently proposed a weighted linear optimization model for multi-criteria ABC inventory classification. Despite its many advantages, Ramanathan’s model (R-model) could lead to a situation where an item with a high value in an unimportant criterion is inappropriately classified as a class A item. In this paper we present an extended version of the R-model for multi-criteria inventory classification. Our model provides a more reasonable and encompassing index since it uses two sets of weights that are most favourable and least favourable for each item. An illustrative example is presented to compare our model and the R-model.     

Review of inventory systems with deterioration since 2001

This paper presents an up-to-date review of the advances made in the field of inventory control of perishable items (deteriorating inventory). The last extensive review on this topic dates back to 2001 (Goyal S.K. and Giri B.C., Recent trends in modeling of deteriorating inventory, European Journal of Operational Research, 134, 1–16). Since then, over two hundred articles on this subject have been published in the major journals on inventory control, indicating the need for a new review. We use the classification of Goyal and Giri based on shelf life characteristics and demand characteristics. Contributions are highlighted by discussing main system characteristics, including price discounts, backordering or lost sales, single or multiple items, one or two warehouses, single or multi-echelon, average cost or discounted cash flow, and payment delay.     

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

Wee and Chung [Wee, H.M., Chung, C.T., 2007. A note on the economic lot size of the integrated vendor–buyer inventory system derived without derivatives. European Journal of Operational Research 177, 1289–1293] use the complete squares method to locate the optimal solution of the integrated system’s total cost TC(Q, B, n). However, their algebra method has shortcomings such that it may be invalid. The main purpose of this paper is to overcome those shortcomings and present complete proofs for Wee and Chung (2007).     

Correct or incorrect application of CAPM? Correct or incorrect decisions with CAPM?

This paper focuses on inconsistencies arising from the use of NPV and CAPM for capital budgeting. It shows that: (i) CAPM capital budgeting decision-making based on disequilibrium NPV is deductively inferred by the capital asset pricing model, (ii) the use of the disequilibrium NPV is widespread in finance both as a decision rule and as a valuation tool, (iii) the disequilibrium NPV does not guarantee additivity nor consistency with arbitrage pricing, so that it is unreliable for valuation, (iv) Magni’s [Magni, C.A., 2002. Investment decisions in the theory of finance: Some antinomies and inconsistencies. European Journal of Operational Research 137, 206–217; Magni, C.A., 2007a. Project valuation and investment decisions: CAPM versus arbitrage. Applied Financial Economics Letters 3 (2), 137–140] criticism of the NPV criterion refers to the disequilibrium NPV, and De Reyck’s [De Reyck, B., 2005. On investment decisions in the theory of finance: Some antinomies and inconsistencies. European of Operational Research 161, 499–504] project valuation method, on the basis of which Magni’s criticism to NPV is objected, leaves decision makers open to arbitrage losses and incorrect decisions.     

Managing supply chain backorders under vendor managed inventory: An incentive approach and empirical analysis

This paper shows how a manufacturer may use an incentive contract with a distributor under a VMI arrangement to gain market share. The manufacturer promises a distributor lower inventory levels in exchange for efforts by the distributor to convert potential lost sales due to stockouts to backorders. Data gathered from a third party provider of information services are then used to illustrate that this incentive arrangement may, at least implicitly, be employed in industry. Our data estimations show that when a manufacturer and distributor are operating under a VMI arrangement, lower inventory at the distributor is associated with a higher conversion rate of lost sales stockouts to backorders.     

Reducing lost-sales rate on the stochastic inventory model with defective goods for the mixtures of distributions

In this paper, we assume that the demands of different customers are not identical in the lead time. Thus, we investigate a continuous review inventory model involving controllable lead time and a random number of defective goods in buyer’s arriving order lot with partial lost sales for the mixtures of distributions of the lead time demand to accommodate more practical features of the real inventory systems. Moreover, we analyze the effects of increasing investment to reduce the lost sales rate when the order quantity, reorder point, lost sales rate and lead time are treated as decision variables. In our studies, we first assume that the lead time demand follows the mixture of normal distributions, and then relax the assumption about the form of the mixture of distribution functions of the lead time demand and apply the minimax distribution free procedure to solve the problem. By analyzing the total expected cost function, we develop an algorithm to obtain the optimal ordering policy and the optimal investment strategy for each case. Finally, we provide numerical examples to illustrate the results.     

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.     

Service constrained inventory models with random lifetimes and lead times

We consider continuous review inventory models with Poisson demands, exponentially distributed lifetimes and replenishment order processing times. The model is unique in that complete backorders, complete lost sales and partial backorders are all addressed. We derive exact expressions of the key operating characteristics. Based on these performance measures, we optimise the system parameters subject to fill rate and waiting time constraints. Numerical examples are also provided to validate our models and to generate useful managerial insights.     

Distribution Free Procedures for Some Inventory Models

Stochastic inventory models, such as continuous review models and periodic review models, require information on the lead time demand. However, information about the form of the probability distribution of the lead time demand is often limited in practice. We relax the assumption that the cumulative distribution function, say F, of the lead time demand is completely known and merely assume that the first two moments of F are known and finte. The minmax distribution free approach for the inventory model consists of finding the most unfavourable distribution for each decision variable and then minimizing over the decision variable. We solve both the continuous review model and the periodic review model with a mixture of backorders and lost sales using the minmax distribution free approach.     

A discussion on “A bivariate optimal replacement policy for a repairable system”

In this short paper, a bivariate optimal replacement policy for a repairable system with a geometric process maintenance model is discussed. Zhang [Zhang, Y.L., 1994. A bivariate optimal replacement policy for a repairable system. Journal of Applied Probability 31, 1123–1127] and Sheu [Sheu, S.H., 1999. Extended optimal replacement model for deteriorating systems. European Journal of Operational Research 112, 503–516] obtained different expression of the long-run average cost per unit time (i.e. average cost rate) of the system respectively. We show that both of their results are correct, therefore, Sheu’s comment (1999) on the result of Zhang (1994) is wrong.     

Some stochastic inventory models with deterministic variable lead time

In this paper we determine optimal reduction in the procurement lead time duration for some stochastic inventory models, jointly with the optimal ordering decisions. The models are developed with complete and partial information about the lead time demand distribution. The stochastic models analyzed in this paper are the classical continuous and periodic review models with a mixture of backorders and lost sales and the base stock model. For each of these models, we provide sufficient conditions for the uniqueness of the optimal operating policy. We also develop algorithms for solving these models and provide illustrative numerical examples.