Two-echelon stochastic inventory system with returns and partial backlogging

This paper investigates a two-echelon stochastic inventory system with returns and partial backlogging; our proposed model, which is realistic for practical situations, might be applicable in many manufacturing planning situations. The objective is to find a continuous-review inventory control policy that minimises the total expected annual cost of the system. A computer code using the software Mathematica 5.2 is developed to derive the optimal solution. Additionally, we discuss the sensitivity of the optimal solution together with the changes of the values of the parameters associated with the model for decision-making. The results are illustrated with the help of four numerical examples.     

A note on inventory replenishment policies for deteriorating items in an exponentially declining market

In determining the replenishment policy for an inventory system, some researchers advocated that the iterative method of Newton could be applied to the derivative of the total cost function in order to get the optimal solution. But, this approach suffers some drawbacks. In this paper we first show the inventory carrying cost is in proportion to the cost of deteriorated items, then offer a formulated approximated solution. Our formula solution is accurate enough to be taken as an optimal solution. Sensitivity analysis shows that our solution is stable. Finally, we give some numerical examples to show the applicability of our solution.Scope and purposeGhare and Schrader initiated the discussion of deteriorating items in the inventory system. After that, many researchers presented diverse models with different assumptions on patterns of deterioration with/without shortages. Demand variation has also impact on inventory policies. In order to determine the optimal lot-size, Wee and Chung & Tsai advocated that the Newton's method is an adequate method. In this paper we illustrate some drawbacks of their approach, mainly the convergence and optimality problem. Thereby, we provide a formulated approximated solution, which is accurate enough to be taken as optimal.     

Coordinating replenishment and pricing policies for non-instantaneous deteriorating items with price-sensitive demand

This article will formulate and solve an inventory system with non-instantaneous deteriorating items and price-sensitive demand. The purpose of this study is to determine the optimal selling price and the length of replenishment cycle such that the total profit per unit time has a maximum value for the retailer. We first establish a proper model for a mathematical formulation. Then we develop several theoretical results and provide the decision-maker with an algorithm to find the optimal solution. Finally, two numerical examples are provided to illustrate the solution procedure, and a sensitivity analysis of the optimal solution with respect to major parameters is carried out.     

An economic production quantity model with random defective rate,rework process and backorders for a single stage production system

This paper revisits the economic production quantity (EPQ) model with rework process at a single-stage manufacturing system with planned backorders. It is well known that any imperfect production system of real life has random defective rates. In this direction, this paper extends an inventory model to allow random defective rates. Basically, three different inventory models are developed for three different distribution density functions such as uniform, triangular, and beta. The analytical derivation provides closed-form solution for each inventory model. We have made comparison tables of optimal results among the distribution functions. Some numerical examples and sensitivity analysis are given to illustrate the inventory models.     

An inventory model with Weibull demand rate,finite rate of production and shortages

We develop here an inventory model with time-dependent two-parameter Weibull demand rate, allowing shortages in the inventory. The shortages are completely backlogged. The production rate is assumed to be finite and proportional to the demand rate. The model is solved analytically to obtain the optimal solution of the problem. It is then illustrated with the help of numerical examples. Sensitivity of the optimal solution with respect to changes in the values of the system parameters is also studied. Special features of time-dependent Weibull demand rate are discussed.     

An EOQ inventory model with time-varying demand and Weibull deterioration with shortages

An inventory model is considered in which inventory is depleted not only by demand, but also by deterioration. Hence, in this paper, we derive the EOQ model for inventory of items that deteriorate at a Weibull-distributed rate, assuming the demand rate with a continuous function of time. Moreover, the proposed model cannot be solved directly in a closed form, thus we used the computer software IMSL MATH/LIBRARY (1989) to find the optimal reorder time. Further, we also find that the optimal procedure is independent from the form of the demand rate. Finally, we also assume that the holding cost is a continuous, non-negative and non-decreasing function of time in order to generalize the EOQ model. Moreover, four numerical examples and sensitivity analyses are provided to assess the solution procedure.     

Optimal replenishment policies with allowable shortages for a product life cycle

In this paper, we investigate replenishment policies with allowable shortages by considering a general, time-varying, continuous, and deterministic demand function for a product life cycle. The objective is to optimally determine the number of inventory replenishments, the inventory replenishment time points, and the beginning time points of shortages within the product life cycle by minimizing the total relevant costs of the inventory replenishment system. The proposed problem is mathematically formulated as a mixed-integer nonlinear programming model. A complete search procedure is developed to find the optimal solution by employing the properties derived in this paper and the well-known Nelder–Mead algorithm. Also, several numerical examples and the corresponding sensitivity analyses are carried out to illustrate the features of our model by utilizing the search procedure developed in this paper.     

A study on an inventory model for non-instantaneous deteriorating items with permissible delay in payments

In this study, an appropriate inventory model for non-instantaneous deteriorating items with permissible delay in payments is considered. The purpose of this study is to find an optimal replenishment policy for minimizing the total relevant inventory cost. This mathematical model is a general framework that comprises numerous previous models such as in Ghare and Schrader [Ghare, P. M., & Schrader, G. H. (1963). A model for exponentially decaying inventory system. International Journal of Production Research, 21, 449–460], Goyal [Goyal, S. K. (1985). Economic order quantity under conditions of permissible delay in payments. Journal of the Operational Research Society, 36, 335–338], and Teng [Teng, J. T. (2002). On the economic order quantity under conditions of permissible delay in payments. Journal of the Operational Research Society, 53, 915–918] as special cases. We have developed some useful theorems to characterize the optimal solutions and provide an easy-to-use method to find the optimal replenishment cycle time and order quantity under various circumstances. Several numerical examples are given to test and verify the theoretical results. Finally, a sensitivity analysis of the optimal solution with respect to major parameters is also included. According to the results of numerical analysis, we provided several ways for the retailer to effectively reduce total annual relevant inventory cost.     

A hub location inventory model for bicycle sharing system design: Formulation and solution

This study addresses a strategic design problem for bicycle sharing systems incorporating bicycle stock considerations. The problem is formulated as a hub location inventory model. The key design decisions considered are: the number and locations of bicycle stations in the system, the creation of bicycle lanes between bicycle stations, the selection of paths of users between origins and destinations, and the inventory levels of sharing bicycles to be held at the bicycle stations. The design decisions are made with consideration for both total cost and service levels (measured both by the availability rate for rental requests at the pick-up rental stations and coverage of the origins and destinations). The optimal design of this system requires an integrated view of the travel costs of users, bicycle inventory costs and facility costs of bicycle stations and bicycle lanes as well as service levels. The purpose of this study is to create a formal model that provides such an integrated view, and to develop methods for obtaining solutions for the design variables in practical situations. The complexity of the problem precludes the exact solution of the optimization problem for instances of realistic size, and so we propose a heuristic method for efficiently finding near-optimal solutions. In the test problem for which enumeration is possible, the heuristic solution is within 2% optimal. Finally, a numerical example is created to illustrate the model and proposed solution algorithm.     

A new algorithm for optimally determining lot-sizing policies for a deteriorating item in an integrated production-inventory system

In this study, we focus on optimally determining lot-sizing policies for a deteriorating item among all the partners in a supply chain system with a single-vendor and multiple-buyers so as to minimize the average total costs. We revise Yang and Wee's [1] model using the Fourier series to precisely estimate the vendor's inventory holding costs. Also, we transform our revised model into a more concise version by applying anapproximation to the exponential terms in the objective function. In order to solve this problem, we analyze the optimality structure of our revised model and derive several interesting properties. By utilizing our theoretical results, we propose a search algorithm that can efficiently solve the optimal solution. Based on our numerical experiments, we show that the proposed algorithm outperforms the existing solution approach in the literature, especially when the number of buyers is larger in the supply chain system.     

Inventory and investment in quality improvement under return on investment maximization

In this paper, we construct and analyze inventory and investment in quality improvement policies under return on investment (ROI) maximization. In our model, the level of quality is represented by the fraction of an order quantity meeting the quality requirements such as product specifications. The key contributions of this paper are the establishment of an ROI model and characterization of the unique global optimal solution. We also show how the inventory level is reduced when it is optimal to invest additional money in quality improvement. In addition, we derive the unique global optimal solutions in closed-form when the investment in quality improvement is a linear function of the quality. Various interesting managerial insights and a numerical example are provided.Scope and purposeQuality improvement and inventory reduction has been extensively studied in accordance with just-in-time (JIT) and/or total quality management (TQM). Meanwhile, for finished products, the return on investment (ROI) is widely used as an economic performance criterion. The purpose of this paper is to design and analyze inventory and investment in quality improvement policies under return on investment (ROI) maximization. Specifically, this paper shows how the level of inventory is reduced quantitatively when it is optimal to invest additional money in quality improvement. In addition, this paper shows how the optimal inventory and investment in quality improvement policies are derived and characterized analytically.     

Optimal ordering policies for deteriorating items using a discounted cash-flow analysis when a trade credit is linked to order quantity

In today’s competitive market, in order to obtain a competition advantage, the supplier often offers the purchaser a longer permissible delay in payments or a price discount if the order quantity is greater than or equal to a predetermined quantity. As a result, in this paper, we establish an inventory model for the purchaser in which the supplier provides different trade credits. We then solve the inventory problem by using a discounted cash-flow (DCF) approach, characterize the optimal solution, and obtain some theoretical results to find the optimal order quantity and the optimal replenishment time. Finally, we provide several numerical examples to illustrate the results.     

An inventory model for ameliorating and deteriorating items taking account of time value of money and finite planning horizon

The objective of this study is to develop an optimal replenishment inventory strategy to consider both ameliorating and deteriorating effects taking account of time value of money and finite planning horizon. The amelioration rate and the deterioration rate are assumed to follow a Weibull distribution. The inventory system is particularly useful for young livestock whose utility increase over time. The discounted cash flow and optimisation technique are used to derive an optimal solution. A numerical example and sensitivity analysis are given to illustrate the theory of the inventory system.     

Warehouse sizing and inventory scheduling for multiple stock-keeping products

In this paper, a commercial problem of enterprise logistics nature called multi-product warehouse sizing problem is formulated and solved. The optimal warehouse inventory level and the order points are determined by minimizing the total inventory ordering and holding costs for a specific time period. The problem is formulated as an appropriate mathematical model of NLP nature and solved using appropriate numerical optimization techniques (successive quadratic programming procedures). The entire mathematical background for formulating the problem and finding the optimal solution to it is presented and appropriately addressed. The model is also illustrated with the help of a numerical case study where the sensitivity of the model parameters involved is given through appropriate figures.     

A supplier selection and order allocation problem with stochastic demands

We consider a system comprising a retailer and a set of candidate suppliers that operates within a finite planning horizon of multiple periods. The retailer replenishes its inventory from the suppliers and satisfies stochastic customer demands. At the beginning of each period, the retailer makes decisions on the replenishment quantity, supplier selection and order allocation among the selected suppliers. An optimisation problem is formulated to minimise the total expected system cost, which includes an outer level stochastic dynamic program for the optimal replenishment quantity and an inner level integer program for supplier selection and order allocation with a given replenishment quantity. For the inner level subproblem, we develop a polynomial algorithm to obtain optimal decisions. For the outer level subproblem, we propose an efficient heuristic for the system with integer-valued inventory, based on the structural properties of the system with real-valued inventory. We investigate the efficiency of the proposed solution approach, as well as the impact of parameters on the optimal replenishment decision with numerical experiments.     

A note on the inventory model for deteriorating items with trapezoidal type demand rate

In this paper, we study the inventory model for deteriorating items with trapezoidal type demand rate, that is, the demand rate is a piecewise linearly function. We proposed an inventory replenishment policy for this type of inventory model. The numerical solution of the model is obtained and also examined.     

Maximizing profits in an inventory model with both demand rate and holding cost per unit time dependent on the stock level

We study an EOQ inventory model with demand rate and holding cost rate per unit time, both potentially dependent on the stock level. The ordering cost, the holding cost and the gross profit from the sale of the item are considered. The objective is to maximize the average profit per unit time. We present the analytical formulation of the problem and demonstrate the existence and uniqueness of the optimal cycle time, giving a numerical algorithm to obtain it. Moreover, we provide two fundamental theoretical results: a rule to check when a given cycle time is the optimal policy, and a necessary and sufficient condition for the profitability of the system. Several EOQ models analyzed by other authors are particular cases of the one here studied. We present some numerical examples to illustrate the proposed algorithm and analyze the sensitivity of the optimal solution with respect to changes in various parameters of the system.     

A comprehensive note on “Lot‐sizing decisions for deteriorating items with two warehouses under an order‐size‐dependent trade credit”

To reduce inventory and increase sales, the supplier frequently offers the retailer a permissible delay in payments if the retailer orders more than or equal to a predetermined quantity. In 2012, Liao et al. proposed an economic order quantity model for a retailer with two warehouses when the supplier offers a permissible delay linked to order quantity. In this paper, we attempt to overcome some shortcomings of their mathematical model. Then, we apply some existing theoretical results in fractional convex programs to prove that the annual total variable cost is pseudoconvex. Hence, the optimal solution exists uniquely, which simplifies the search for the global minimum solution to a local minimum solution. Finally, we run a couple of numerical examples to illustrate the problem and compare the optimal solutions between theirs and ours.     

A replenishment policy for items with price-dependent demand,time-proportional deterioration and no shortages

In this article, an order-level inventory system for deteriorating items has been developed with demand rate as a function of selling price. The demand and the deterioration rate are price dependent and time proportional, respectively. We have considered a perishable item that follows a three-parameter Weibull distribution deterioration. Shortages are not permitted in our model. The optimal solution is illustrated with a numerical example and the sensitivity analysis of parameters is carried out.     

Deterministic inventory model for deteriorating items with trade credit financing and capacity constraints

This work presents an inventory model for optimizing the replenishment cycle time for a single deteriorating item under a permissible delay in payments and constraints on warehouse capacity (owned warehouse capacity, with excess inventory stored in rental warehouses). Rented warehouses are assumed to charge higher unit holding costs than owned warehouses. Furthermore, item deterioration rates are assumed to differ between warehouses. This study has two main purposes: First, the mathematical models of the inventory system are established under the above conditions. Second, this study demonstrates that the optimal solution not only exists but is unique, and two theorems are devised for determining the optimal replenishment cycle time. Finally, numerical examples are presented to illustrate the resulting theorems.