Long-run Average Cost Function Research Articles

The residual time approach for (Q, r) model under perishability, general lead times, and lost sales

We consider a (Q, r) perishable inventory system with state-dependent compound Poisson demands with a random batch size, general lead times, exponential shelf times, and lost sales. We assume $$r<Q$$ and analyze the system using an embedded Markov process at the replenishment points. Using the queueing and Markov chain decomposition approach, we characterize the distribution of the residual lead time and derive the stationary distribution of the inventory level. We construct closed-form expressions for the expected total long-run average cost function. The closed form allows us to efficiently obtain, numerically, the optimal Q and r parameters. Numerical study provides several guidelines for the optimal control. For example, we show that approximating the lead time distribution by an exponential one only works when the optimal reorder point of the approximation is very small; in other cases the usage of the exact distribution can lead to substantial cost savings (up to 14%). We further provide intuition insight on the optimal controls and how they depend on different factors, e.g., the lead time variability, and the demand features (arrival rate, size and variability).

A state-dependent perishability (s, S) inventory model with random batch demands

We consider continuous-review perishable inventory models with random lead times and state-dependent Poisson demand. The paper revises an earlier work of Barron and Baron (IISE Trans 1–52, 2019). While the former studies unit Poisson demands, this paper deals with demand uncertainty and allows for random batch demands. We conduct a comprehensive analysis of two main models that have different lead times and perish times under backorders or lost sales. Thus, our models can be applied to many industries, in situations where the system is subject to random perishability, random lead time, and demand uncertainty. With a probabilistic approach, we derive a long-run average cost function under the (S, s) replenishment policy. Numerical examples are used to demonstrate the impact of changing batch size and other system parameters on the optimal policy. Our numerical study indicates that, although the Markovian policy can be used as a good approximation of the average total cost, it performs better for a general perish time. We further show that the optimal cost may differ for a different average batch size, while the batch variability seems to provide some robustness.

Mathematical modelling for determining the replenishment policy for deteriorating items in an EPQ model with multiple shipments

ABSTRACTThis paper extends an economic production quantity (EPQ) model for deteriorating items when the delivery policy is based on multiple shipments. A continuous inventory policy for satisfying demand has been considered in the classic EPQ model. While in an integrated production–inventory–delivery system, the multi-shipment approach is actually used instead of the continuous issuing policy. On the other hand, the EPQ model has not been developed for deteriorating items with multiple shipments. This paper considers the time-varying deterioration rate and finds optimal replenishment lot size which minimises the long-run average cost function after proving its convexity. Finally, mathematical modelling and analysis are employed for a laboratory kit production system. Then, impact factors which can modify total cost are recognised and optimal replenishment lot size, as well as optimal quantity of each shipment, is found in different situations.

Setup cost reduction EMQ inventory system with probabilistic defective and rework in multiple shipments management

This paper considers an economic manufacturing quantity (EMQ) model for defective products with imperfect production processes and rework, in which the setup cost is logarithmic function of capital investment. In this study, fixed quantity multiple installments of the finished batch are delivered to customers at a fixed interval of time. We also consider three types of continuous probabilistic defective function to find the associated cost. The way our basic business operations like decision making, marketing strategies, financial management, etc. are done are being reformed with the use of computers and mathematics. In view of that the mathematical modeling and computational algorithm are employed in this study for optimizing the replenishment lot-size and setup cost simultaneously with the objective of minimizing total cost of the EMQ system. The long-run average cost function is derived, its convexity is proved via differential calculus. A computer code using the software Matlab is developed to derive the optimal solution and present numerical examples to illustrate the model.

An EMQ inventory model for defective products involving rework and sales team’s initiatives-dependent demand

This paper investigates the issue of an economic manufacturing quantity model for defective products involving imperfect production processes and rework. We consider that the demand is sensitive to promotional efforts/sales teams’ initiatives as well as the setup cost can be reduced through further investment. It also assumes that fixed quantity multiple installments of the finished batch are delivered to customers at a fixed interval of time. The long-run average cost function is derived and its convexity is proved via differential calculus. An effective iterative solution procedure is developed to achieve optimal replenishment lot-size, setup cost and the initiatives of sales teams so that the total cost of system is minimized. Numerical and sensitivity analyses are performed to evaluate the outcome of the proposed solution procedure presented in this research.

One for one period policy for perishable inventory

Recently, for zero ordering cost a new ordering policy named (1, T), in which the time interval between two consecutive orders and the value of the order size are both constant, have been developed for nonperishable products. In this paper, the (1, T) policy is developed for perishable products. Using an analogy among this inventory model, a queueing model with impatient customers, and a finite dam model, the long-run average total cost function of the inventory system is derived. It is observed that the total cost rate is independent from the lead time as is for nonperishable products. Since analyzing the convexity of the model is extremely complicated, a proposition is proved to define a domain for the optimal solution, and then a search algorithm is presented to obtain the optimal solution. Furthermore, a numerical analysis is carried out to examine the sensitivity of optimal T with respect to system parameters and to compare the performance of (1, T) policy with the well known (S−1, S) policy. This analysis shows that for fixed values of system parameters, there is a fixed value of lead time for which the performance of (1, T) policy is better than (S−1, S) policy. Further as the lead time increases this superiority is more pronounced.

Remarks on the optimization method of a manufacturing system with stochastic breakdown and rework process in supply chain management

Chiu et al. (2010) [8] present the proof of convexity of the long-run average cost function E[TCU(t1)] for a manufacturing system with stochastic breakdown and rework process. This note not only demonstrates that E[TCU(t1)] is not convex but also adopts the rigorous methods of mathematics to develop the complete solution procedure to find the optimal solution for removing shortcomings of the above paper mentioned.

Flexible service policies for a Markov inventory system with two demand classes

This paper explores flexible service policies for an (r, Q) Markov inventory system with two classes of customers, ordinary and prioritized customers. When the on-hand inventory drops to pre-determined safety level r, arrival ordinary customers receive service at probability p. Firstly, the inventory level state transitions equation is set up. The steady-state probability distribution and the system's performance measures which are used for the inventory control are derived. Next, a long-run average inventory cost function is established and a mixed integer optimization model is set up. And, an improved genetic algorithm for the optimum control policies is presented. Finally, the optimal inventory control polices and the sensitivities are investigated through numerical experiments.

Analysis of an assemble-to-order system with different review periods

We consider a single item assembled from two components. One of the components has a long lead time, high holding cost and short review period as compared to the other one. We assume that net stocks are reviewed periodically, customer demand is stochastic and unsatisfied demand is backordered. We analyze the system under two different policies and show how to determine the policy parameters that minimize average holding and backorder costs. First, we consider a pure base stock policy, where orders for each component are placed such that the inventory position is raised up to a given base stock level. In contrast to this, only the orders for one component follow this logic while the other orders are synchronized in case of a balanced base stock policy. Through mathematical analysis, we come up with the exact long-run average cost function and we show the optimality conditions for both policies. In a numerical study the policies are compared.

Solving production lot size problem with an improved delivery policy and failure in rework by a simplifi ed solution procedure

A simplifi ed solution procedure is proposed in this paper to solve a specifi c production lot size problem with an improved delivery policy and failure in rework. Unlike the conventional method using differential calculus with the need for proving optimality of the longrun average cost function, this paper demonstrates that the optimal lot size and its related costs for the aforementioned production model can be derived without derivatives. Such an approach may enable practitioners who with little knowledge of calculus to understand with ease the realistic production-shipment integrated systems

Alternative approach for solving replenishment lot size problem with discontinuous issuing policy and rework

Conventional approach for solving the replenishment lot size problem is by using differential calculus on the long-run average production cost function with the need to prove optimality first. Recent studies proposed an algebraic approach to the solution of classic economic order quantity (EOQ) and the economic production quantity (EPQ) models without reference to the use of derivatives. This paper extends it to the solution of a specific EPQ model as was examined by Chiu et al. [Chiu, S. W, Chen, K. -K, Lin, H. -D. Numerical method for determination of the optimal lot size for amanufacturing system with discontinuous issuing policy and rework. International Journal for Numerical Methods in Biomedical Engineering. doi: 10.1002/cnm.1369. (in Press; Published online March-10-2010).]. As a result, optimal replenishment lot size and a simplified optimal production-inventory cost formula for such a particular EPQ model can be derived without derivatives. This alternative approach may enable practitioners who with little knowledge of calculus to understand the realistic production systems with ease.

A study on improving the optimization process of a manufacturing run time problem with defective and machine breakdown

This research note improves the optimization process of a manufacturing run time problem studied by Chiu et al. (2011) by presenting a direct proof of convexity of the long-run average cost function of such a problem. It can be used to replace Theorem 1 on conditional convexity given by Chiu et al. enhance quality of optimization process, and eliminate the computational efforts in verifying the conditional convexity. Key words: Optimization process, convexity, manufacturing, replenishment run time, breakdown, production control.

Mathematical modeling for determining the replenishment policy for EMQ model with rework and multiple shipments

This paper uses mathematical modeling to determine optimal inventory replenishment policy for the economic manufacturing quantity (EMQ) with rework and multiple shipments. The classic EMQ model considers a continuous inventory issuing policy for satisfying customer’s demands. It also assumes that all products produced are of perfect quality. However, in a real world vendor-buyer integrated environment, multi-shipment policy is practically used in lieu of the continuous issuing approach and generation of random defective items is inevitable. In this study, we assume the reworking of all defective items takes place when the regular production process ends in each cycle. Failure in repair exists; a portion of reworked items fails during the reworking and becomes scrap. The finished items can only be delivered to customers if the whole lot is quality assured at the end of rework. Fixed quantity multiple installments of the finished batch are delivered to customers at a fixed interval of time. Mathematical modeling and analysis are employed in this study for solving such a realistic EMQ model. The long-run average cost function is derived, its convexity is proved, and a closed-form optimal manufacturing lot size is obtained. Two special cases to the proposed model are examined.

Production inventory system with random supply interruptions statue and random lead times

This article analyzes a continuous-review inventory system with random supply interruptions and random lead time which may be interrupted by a random number of supplier's OFF periods. The inventory with constant demand rate is managed by a ( r;q1,q2,…,qm) policy and supplies from an unreliable sole supplier. By renewal theory and matrix Geometric method, the long-run average cost function is obtained and some important properties of the function are proved. Furthermore, performance of the inventory is derived.

An Algebraic Approach for Determining the Optimal Lot Size for EPQ Model with Rework Process

This paper presents a simple algebraic approach for deriving the optimal lot size for economic production quantity (EPQ) model with rework process. Conventional methods for solving lot size problems are by using differential calculus on the long-run average production-inventory cost function with the need to prove optimality first. A few recent articles proposed the algebraic approach to the solution of classic economic order quantity (EOQ) and EPQ model without reference to the use of derivatives. This paper extends it to an EPQ model with reworking of defective items. We demonstrate that optimal lot size and optimal production-inventory cost for such an imperfect EPQ model can be derived without derivatives. As a result, it may enable the practitioners or students who with little knowledge of calculus to understand or handle with ease the realistic production systems.

A centralized ordering and allocation system with backorders and strategic lost sales

This article considers a multi-retailer distribution system that is managed by a central decision maker who, at the start of each period, determines how much to order to replenish the system stock. The decision maker also determines how to allocate incoming pipeline inventory to maintain inventory balance among the retailers. It has been noted in the literature that balancing inventories can equalize service levels among retailers. To improve the efficacy of the allocation, this article allows some demand to be rejected to keep inventories in balance. Consequently, depending on the realized pattern of demand during the delivery lead time, the inventory is dynamically allocated to each of the retailers. For the model with two retailers, an exact representation of the infinite-horizon long-run average cost function is developed. This exact expression is used to develop conditions for the unique solution for the two-retailer case. The presented analysis holds for a wide class of continuously and discretely distributed demand.

Remarks on the optimization process of a manufacturing system with stochastic breakdown and rework

This paper reexamines the optimization process of a manufacturing system with stochastic breakdown and rework proposed by Chiu [S.W. Chiu, An optimization problem of manufacturing systems with stochastic machine breakdown and rework process, Applied Stochastic Models in Business and Industry 24 (2008) 203–219]. The proof of convexity of the long-run average cost function for the aforementioned manufacturing system is provided in this note. It can be used to replace the conditional convexity given in Theorem 1 of Chiu (2008) [1]. Therefore, when determining the optimal solution for such a real-life system, computational efforts in verifying the conditional convexity can now be omitted, due to the improved quality of the optimization process.

Numerical method for determination of the optimal lot size for a manufacturing system with discontinuous issuing policy and rework

This paper employs numerical method for determination of the optimal lot size for a manufacturing system with discontinuous inventory issuing policy and imperfect rework of random defective items. The classic economic manufacturing quantity (EMQ) model assumes a continuous issuing policy for satisfying customer's demands, and perfect quality production for all items produced during a production run. However, in a real-life vendor–buyer integrated system, the discontinuous issuing policy such as multi-shipment policy is practically used in lieu of continuous issuing policy, and it is inevitable to generate defective items during a production run. Imperfect quality items fall into two groups: the scrap and the reworkable. During the rework process, failure in repair exists; a portion of reworked items fails and becomes scrap. The finished items can only be delivered to customers if the whole lot is quality assured at the end of the rework. Mathematical modeling and analysis are used to deal with the proposed model, and the long-run average cost function is derived. Convexity of this cost function is proved and a closed-form optimal batch size solution to the problem is obtained. Two special cases are examined and a numerical example demonstrates its practical usage. Copyright © 2010 John Wiley & Sons, Ltd.

THE OPTIMAL INVENTORY CONTROL OF A SYSTEM WITH RANDOM LEAD TIME WHERE RANDOM SUPPLY INTERRUPTIONS AFFECT THE REPLENISHMENT

This paper analyzes a continuous-review inventory system with random supply interruptions and stochastic lead time. In such a system, a monopolistic supplier alternates between random ON and OFF periods. The variable (r,q)-policy is used. With a constant demand rate, the leadtime may be interrupted by a random number of OFF periods of the supplier. Using the regenerative cycle analysis, a long-run average cost function is obtained and shown to have some important properties. Accordingly, an effective search algorithm is proposed to find the sub-optimal inventory policy. In the end, we also investigate the effects on the average cost of the uncertain order delay which attribute to both the supply interruptions and the random leadtime. Numerical examples are presented to demonstrate the effectiveness of the search algorithm.

Optimal replenishment policy for manufacturing systems with failure in rework, backlogging and random breakdown

This article studies the optimal replenishment policy for an economic manufacturing quantity (EMQ) model with failure in rework, backlogging and random breakdown. A recent article [P.A. Hayek and M.K. Salameh, Prod. Plann. Control 2001;12:584–590] investigated the optimal lot size problem on an imperfect quality EMQ model. However, in most real-life manufacturing systems random breakdown is another inevitable reliability factor. To cope with the stochastic machine failures, the production planners need to calculate the mean time between failures and establish a robust plan accordingly, so that the overall production-inventory costs for such an unreliable system can be minimized. This study incorporates random machine breakdown and failure in rework factors into the imperfect EMQ model examined by the aforementioned paper. Mathematical modelling and cost analysis are employed and the renewal reward theorem is used to deal with variable cycle length. Convexity of the long-run average cost function is proved and optimal replenishment policy for such an imperfect EMQ model is derived. Numerical example demonstrates its practical usages.