A continuous review inventory model with controllable backorder rate and investments

In this study, we investigate a continuous review inventory model to reduce lead time, yield variability and setup cost simultaneously through capital investments. We assume that the backorder rate is depending on the length of the lead time through the amount of shortages. We also assume that lead time demand's distribution is not known but its first and second moments are known. We apply minimax distribution free procedure to minimise the expected total annual cost. By using logarithmic investment function we describe the relationship between the reduction in lead time, yield variability and setup cost with capital investment. This function was used in many existing models. Our main aim is to determine the optimal capital investment and ordering policies that minimises the expected total annual cost. To find out the optimal solution, an algorithm is given. With the help of this algorithm, optimal capital investment and ordering policies are wrought out. Numerical examples are given to elucidate the model. Our proposed model greatly differs from the model existing in the literature (the model by Lin and Hou (2005 Lin, LC and Hou, KL. 2005. An Inventory System with Investment to Reduce Yield Variability and Setup Cost. Journal of the Operational Research Society, 56: 67–74. [Taylor &; Francis Online], [Web of Science ®] , [Google Scholar])) viz: (1) In the above model, yield variability and setup cost were reduced through capital investment. In our model we reduce yield variability setup cost and also the lead time, which plays a vital role in any business. By reducing lead time we can improve the service level to the customer so as to increase the competitive edge in business. (2) In the model (the model by Lin and Hou (2005 Lin, LC and Hou, KL. 2005. An Inventory System with Investment to Reduce Yield Variability and Setup Cost. Journal of the Operational Research Society, 56: 67–74. [Taylor &; Francis Online], [Web of Science ®] , [Google Scholar])), it was assumed that lead time demand follows normal distribution. But in our model we take the distribution of lead time demand as distribution free. That is, it can follow any distribution which is more general. (3) In the above model (the model by Lin and Hou (2005 Lin, LC and Hou, KL. 2005. An Inventory System with Investment to Reduce Yield Variability and Setup Cost. Journal of the Operational Research Society, 56: 67–74. [Taylor &; Francis Online], [Web of Science ®] , [Google Scholar])), shortages are completely backlogged. But we consider partial backlogging and take the backlogging rate as 0 ≤ B ≤ 1. If we set backlogging rate B = 1 we get the above model. That is, the above model is particular case of our model. (4) We also assume that the backorder rate depends on the length of the lead time through the amount of shortages. If the lead time is longer then shortage accumulation is higher. The patience of customers will result in failure in business since some customers may turn to some other supplier. Hence, the backorder rate will be reduced. This assumption is very realistic.     

A two-stage hybrid manufacturing model with controllable make-to-order production rates

As an effective strategy to facilitate delivering customized products within short lead time, hybrid manufacturing via a two-stage process has received attention from academia and industry. In this paper, we study a two-stage hybrid manufacturing system in which semifinished products are manufactured in a make-to-stock fashion in the first stage and end-products are produced from semifinished goods in a make-to-order (MTO) mode in the second stage. The rate of MTO production can be controlled within given limits, depending on the status of the system. The primary goal of this paper is to study a policy for coordinating order admission, MTO production rate, and inventory replenishment controls. Formulating the problem as a Markov decision process model, we characterize the structure of optimal control policies to maximize the long-run average profit. Using a numerical experiment, we study how the flexibility in MTO production rate affects the optimal policy and the optimal profit. We also examine the effect of the number of alternative MTO production rates on the optimal profit. We propose three heuristic policies implementable for general cases. The first heuristic describes two linear switching functions for admission and production controls and a selection rule for MTO production rate control. The second heuristic specifies fixed thresholds for the control decisions using the local information. The third heuristic presents linear switching functions that approximate the optimal threshold curves. Unlike second and third heuristics, the first heuristic does not require a grid search to determine the control parameters. We implement numerical studies to examine the marginal impact of system parameters and the effect of the number of alternative MTO production rates on the performance of the heuristics. Compared to the optimal policy, the average percentage performance of the first and third heuristics is less than 1% for both numerical studies. On the other hand, the average percentage performance of the second heuristic is larger than 3%, and it exceeds 10% for a set of particular problem examples.     

Analysis of a dynamic lot‐sizing problem with production capacity constraint

We study a capacitated dynamic lot‐sizing problem with special cost structure involving setup cost, freight cost, production cost, and inventory holding cost. We investigate two cases of the problem categorized by whether the maximal production capacity in one period is an integral multiple of the capacity of a container and reveal the special structure of an optimal solution for each case. In the case where the maximal production capacity is an integral multiple of a container's capacity, the T‐period problem is solved using polynomial effort by a network algorithm. For the other case, the problem is transformed into a shortest path problem, and a network‐based algorithm combining dynamic programming is proposed to solve it in polynomial time. Numerical examples are presented to illustrate application of the algorithms to solve the two cases of the problem.     

Economic production quantity model with a shifting production rate

Typical models for determining the economic production quantity (EPQ) assume perfect product quality and perfect production processes. Deteriorating processes may affect production systems in several ways. They may decrease the quality of the items produced, cause production stoppage and breakdowns and/or reduce the production rate due to production process inefficiency. The purpose of this paper is to present an EPQ model that incorporates the effect of shifts in production rate on lot sizing decisions due to speed losses. The cycle starts with a certain production rate and after a random time, the production rate shifts to a lower value. A mathematical model to determine the optimal production policy under these conditions is developed and analyzed. Numerical examples are presented for illustrative purposes.     

Optimal lot sizing with maintenance actions and imperfect production processes

Porteus (1986) explored an economic order quantity model with imperfect production processes that the approximate lot size is derived. Basically, he dealt with the lot size problem is rather meaningful. However, for mathematical simplicity, he adopted a truncated Taylor series expansion to present the approximate expected total cost function that results in overvalue of expected total cost. In this paper, we extend Porteus (1986) to present the optimal lot size model for defective items with a constant probability when the system is out-of-control and taking the maintenance cost into account. We show that there exists a unique optimal lot size such that the expected total cost is minimised. In addition, the bounds of optimal lot size are provided to develop the solution procedure. Finally, numerical examples are given to illustrate the theoretical results and compare optimal solutions obtained by using our approach and Porteus's approach. Numerical results show that our approach is better.     

Mixture inventory model involving variable lead time and controllable backorder rate

This paper allows the backorder rate as a control variable to widen applications of Ouyang et al.'s model [J. Oper. Res. Soc. 47 (1996) 829]. In this study, we assume that the backorder rate is dependent on the length of lead time through the amount of shortages. We discuss two models that are perfect and partial information about the lead time demand distribution, that is, we first assume that the lead time demand follows a normal distribution, and then remove this assumption by only assuming that the first and second moments of the probability distribution of lead time demand are known. For each case, we develop an algorithm to find the optimal ordering strategy. Three numerical examples are given to illustrate solution procedure.     

A numerical solution for a two-stage production and inventory system with random demand arrivals

This paper is about the study of a production lot sizing problem consisting of customers, one retailer, and one manufacturer. Demand from customers arrives randomly at a retailer one unit at a time. The retailer replenishes inventory from the manufacturer upon receiving a customer's order after its inventory depleted to zero. The manufacturer's production rate is assumed to be a finite constant. A production cycle starts when the manufacturer's inventory falls to or below zero and stops when its on-hand inventory reaches its optimal level. During the uptime in a production cycle, inventory is being built while randomly arriving orders from retailer are being fulfilled. The order arrival times from customers are independently and identically distributed, hence the inventory processes at both the manufacturer and the retailer become a renewal process that is difficult to solve analytically for a general distribution of order arrival time. Therefore, a numerical approach is used in developing a search procedure to obtain the optimal solution to the problem. Employing such a numerical approach, we also investigate how optimal solutions in different cases will change over the spectrum of some key parameters of the problem.     

Economic lot sizing in a production system with random demand

An extended economic production quantity model that copes with random demand is developed in this paper. A unique feature of the proposed study is the consideration of transient shortage during the production stage, which has not been explicitly analysed in existing literature. The considered costs include set-up cost for the batch production, inventory carrying cost during the production and depletion stages in one replenishment cycle, and shortage cost when demand cannot be satisfied from the shop floor immediately. Based on renewal reward process, a per-unit-time expected cost model is developed and analysed. Under some mild condition, it can be shown that the approximate cost function is convex. Computational experiments have demonstrated that the average reduction in total cost is significant when the proposed lot sizing policy is compared with those with deterministic demand.     

Production/inventory systems with a stochastic production rate under a continuous review policy

A single production facility is dedicated to producing one product with completed units going directly into inventory. The unit production time is a random variable. The demand for the product is given by a Poisson process and is supplied directly from inventory when available, or is backordered until it is produced by the production facility. Relevant costs are a linear inventory holding cost, a linear backorder cost, and a fixed setup cost for initiating a production run. The objective is to find a control policy that minimizes the expected cost per time unit.The problem may be modeled as an M/G/1 queueing system, for which the optimal decision policy is a two-critical-number policy. Cost expressions are derived as functions of the policy parameters, and based on convexity properties of these cost expressions, an efficient search procedure is proposed for finding the optimal policy. Computational test results demonstrating the efficiency of the search procedure and the behavior of the optimal policy are presented.     

Computational procedure of optimal inventory model involving controllable backorder rate and variable lead time with defective units

This article considers that the number of defective units in an arrival order is a binominal random variable. We derive a modified mixture inventory model with backorders and lost sales, in which the order quantity and lead time are decision variables. In our studies, we also assume that the backorder rate is dependent on the length of lead time through the amount of shortages and let the backorder rate be a control variable. In addition, we assume that the lead time demand follows a 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. Furthermore, we develop an algorithm procedure to obtain the optimal ordering strategy for each case. Finally, three numerical examples are also given to illustrate the results.     

A mixed inventory model with variable lead time and random back-order rate

In this paper, we study the determination of the optimal lead time, reorder point and order quantity considering that the back-order probability of a demand made during a stock-out period depends on the interval from the moment in which the order is placed until the next replenishment. Two models are analysed for the specification of the back-order probability: exponential functions and piecewise constant functions. The distribution of the lead time demand is assumed to be Poisson. An algorithm for the determination of the optimal order quantity, reorder point and lead time is given. A numerical example is presented to illustrate the results.     

A note on defective units in an inventory model with sub-lot sampling inspection for variable lead-time demand with the mixture of free distributions

In a recent paper Wu and Ouyang (2000) assumed that an arriving order lot may contain some defective items and considered that the number of defective items in the sub‐lot sampled to be a random variable. They derived a modified mixture inventory model with backorders and lost sales, in which the order quantity, re‐order point, and the lead‐time were decision variables. In their studies they assumed that the lead‐time demand followed a normal distribution for the first model and relaxed the assumption about the form of the distribution function of the lead‐time demand for the second model. When the demand of the different customers is not identical with regard to the lead‐time, then one cannot use only a single distribution (such as Wu and Ouyang (2000) ) to describe the demand of the lead‐time. Hence, we extend and correct the model of Wu and Ouyang (2000) by considering the lead‐time demand with the mixed normal distributions (see Everitt and Hand (1981) , and Wu and Tsai (2001) ) for the first model and the lead‐time demand with the mixed distributions for the second model. And we also apply the minimax mixed distributions free approach to the second model. Moreover, we also develop an algorithm procedure to obtain the optimal ordering strategy for each case.     

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.     

可控前置时间的连续盘点联合库存策略研究

鉴于控制前置时间对精益生产系统的重要性,在考虑买方与卖方合作的同时,扩展Goyal生产批量交货的假设,假设需求服从正态分布,以订购数量、运送次数与前置时间为决策变量,建立前置时间可控制的联合库存模型以确定适当的库存水平,使得库存总成本最小化,且可以通过协商在买卖双方之间进行节省成本的分配。进行了数值范例,并将联合库存模型与Banerjee模型、Goyal模型进行了比较。     

A bi-objective continuous review inventory control model: Pareto-based meta-heuristic algorithms

In this paper, a bi-objective multi-product (r,Q) inventory model in which the inventory level is reviewed continuously is proposed. The aim of this work is to find the optimal value for both order quantity and reorder point through minimizing the total cost and maximizing the service level of the proposed model simultaneously. It is assumed that shortage could occur and unsatisfied demand could be backordered, too. There is a budget limitation and storage space constraint in the model. With regard to complexity of the proposed model, several Pareto-based meta-heuristic approaches such as multi-objective vibration damping optimization (MOVDO), multi-objective imperialist competitive algorithm (MOICA), multi-objective particle swarm optimization (MOPSO), non-dominated ranked genetic algorithm (NRGA), and non-dominated sorting genetic algorithm (NSGA-II) are applied to solve the model. In order to compare the results, several numerical examples are generated and then the algorithms were analyzed statistically and graphically.     

A production/remanufacturing inventory model with price and quality dependant return rate

Inventory management of produced, remanufactured/repaired and returned items has been receiving increasing attention in recent years. The available studies in the literature consider a production environment that consists of two shops. The first shop is for production and remanufacturing/repair, while the second shop is for collecting used (returned) items to be remanufactured in the first shop, where demand is satisfied from producing new and from remanufacturing/repairing returned items. Numerical and analytical results from these developed models suggested that a pure (bang–bang) policy of either no waste disposal (total remanufacturing) or no remanufacturing (pure production and total disposal) is the best strategy, while the mixed strategy (a mixture of production and remanufacturing) is the optimum case under certain limited assumptions. In practice, the quality of the returned items and the purchasing price that reflects this quality is what usually governs a collection (or return) policy of used items. Unlike those available models in the literature, this paper suggests that the flow of returned items is variable, and is controlled by two decision variables, which are the purchasing price for returned items corresponding to an acceptance quality level. Deterministic mathematical models are presented for multiple remanufacturing and production cycles.     

Computational aspects of an extended EMQ model with variable production rate

The paper considers a generalized economic manufacturing quantity (EMQ) model with stochastic machine breakdown and repair in which the time to machine failure, corrective and preventive repair times are all assumed to be random variables. The model is formulated under general failure and general repair time distributions, treating the machine production rate (speed) as a decision variable. As the stress condition of the machine changes with the production rate, the failure rate is assumed to be dependent on the production rate. The model is further extended to the case where certain safety stocks are hold in inventory to protect against possible stockout during machine repair. The solution procedure and computational algorithms of the associated constrained optimization problems are provided. Numerical examples are taken to determine the optimal production policies by the proposed algorithms and examine the sensitivity of the model parameters.Several economic manufacturing quantity (EMQ) models for unreliable manufacturing systems have been developed in the literature even for general failure and general repair (corrective) time distributions. In these studies, preventive repair has not been considered in a general way and efforts have been made to derive the production control and maintenance policy for inflexible manufacturing systems, where the machine capacity is pre-determined. The purpose of this article is to formulate a generalized EMQ model for a flexible unreliable manufacturing system in which (i) the time to machine failure and repair (corrective and preventive) times follow general probability distributions and (ii) the machine failure rate is dependent on the production rate. Consideration of a variable production rate makes the model hard to analyze completely. So, attempt has also been made to get into its computational aspects by developing solution algorithms.     

Continuous review inventory model with controllable lead time,lost sales rate and order processing cost when the received quantity is uncertain

In practice the quantity received may not match the quantity ordered due to worker's strike, rejection during inspection, damage during transportation, human errors in counting, etc. Accordingly, the managers often must make decisions under uncertain quantity received circumstances. In this study, we investigate the continuous review inventory model with shortages include the case where the quantity received is uncertain, in which the lead time, lost sales rate and order processing cost are decision variables. Here, we consider the lead time crashing cost is an exponential function of lead time, and the order processing cost and lost sales rate are logarithmic functions of capital investment. The objective of this study is to minimize the total relevant cost by simultaneously optimizing the order quantity, lost sales rate and order processing cost. In addition, an efficient algorithm is developed to determine the optimal policy, and our approach is illustrated through a numerical example. From the results of numerical example, it can be shown that, the significant savings can be achieved through the reductions of order processing cost and lost sales rate.     

An inventory model for deteriorating items under stock-dependent demand and controllable deterioration rate

In this paper, we formulate a deteriorating inventory model with stock-dependent demand by allowing preservation technology cost as a decision variable in conjunction with replacement policy. Moreover, it is assumed that the shortages are allowed and partially backlogged, depending on the length of the waiting time for the next replenishment. The objective is to find the optimal replenishment and preservation technology investment strategies while maximizing the total profit per unit time. For any given preservation technology cost, we first prove that the optimal replenishment schedule not only exists but is unique. Next, we show that the total profit per unit time is a concave function of preservation technology cost when the replenishment schedule is given. We then provide a simple algorithm to find the optimal preservation technology cost and replenishment schedule for the proposed model. Finally, we use some numerical examples to illustrate the model.     

A hybrid credibility-based fuzzy multiple objective optimisation to differential pricing and inventory policies with arbitrage consideration

In most markets, price differentiation mechanisms enable manufacturers to offer different prices for their products or services in different customer segments; however, the perfect price discrimination is usually impossible for manufacturers. The importance of accounting for uncertainty in such environments spurs an interest to develop appropriate decision-making tools to deal with uncertain and ill-defined parameters in joint pricing and lot-sizing problems. This paper proposes a hybrid bi-objective credibility-based fuzzy optimisation model including both quantitative and qualitative objectives to cope with these issues. Taking marketing and lot-sizing decisions into account simultaneously, the model aims to maximise the total profit of manufacturer and to improve service aspects of retailing simultaneously to set different prices with arbitrage consideration. After applying appropriate strategies to defuzzify the original model, the resulting non-linear multi-objective crisp model is then solved by a fuzzy goal programming method. An efficient stochastic search procedure using particle swarm optimisation is also proposed to solve the non-linear crisp model.