On the optimal control of manufacturing and remanufacturing activities with a single shared server

We consider a single-product make-to-stock manufacturing–remanufacturing system. Returned products require remanufacturing before they can be sold. The manufacturing and remanufacturing operations are executed by the same single server, where switching from one activity to another does not involve time or cost and can be done at an arbitrary moment in time. Customer demand can be fulfilled by either newly manufactured or remanufactured products. The times for manufacturing and remanufacturing a product are exponentially distributed. Demand and used products arrive via mutually independent Poisson processes. Disposal of products is not allowed and all used products that are returned have to be accepted. Using Markov decision processes, we investigate the optimal manufacture–remanufacture policy that minimizes holding, backorder, manufacturing and remanufacturing costs per unit of time over an infinite horizon. For a subset of system parameter values we are able to completely characterize the optimal continuous-review dynamic preemptive policy. We provide an efficient algorithm based on quasi-birth–death processes to compute the optimal policy parameter values. For other sets of system parameter values, we present some structural properties and insights related to the optimal policy and the performance of some simple threshold policies.     

Determining the fill rate for a periodic review inventory policy with capacitated replenishments,lost sales and zero lead time

In this paper we consider a periodic review order-up-to inventory system with capacitated replenishments, lost sales and zero lead time. We consider discrete demand. It is shown that the initial stock levels of the different review periods form a Markov chain and we determine the transition matrix. Furthermore we study for what probability mass functions of the review period demand the Markov chain has a unique stationary distribution. Finally, we present a method to determine the fill rate.     

On the exact calculation of the fill rate in a periodic review inventory policy under discrete demand patterns

The primary goal of this paper is the development of a generalized method to compute the fill rate for any discrete demand distribution in a periodic review policy. The fill rate is defined as the fraction of demand that is satisfied directly from shelf. In the majority of related work, this service metric is computed by using what is known as the traditional approximation, which calculates the fill rate as the complement of the quotient between the expected unfulfilled demand and the expected demand per replenishment cycle, instead of focusing on the expected fraction of fulfilled demand. This paper shows the systematic underestimation of the fill rate when the traditional approximation is used, and revises both the foundations of the traditional approach and the definition of fill rate itself. As a result, this paper presents the following main contributions: (i) a new exact procedure to compute the traditional approximation for any discrete demand distribution; (ii) a more suitable definition of the fill rate in order to ignore those cycles without demand; and (iii) a new standard procedure to compute the fill rate that outperforms previous approaches, especially when the probability of zero demand is substantial. This paper focuses on the traditional periodic review, order up to level system under any uncorrelated, discrete and stationary demand pattern for the lost sales scenario.     

Fill rate of single-stage general periodic review inventory systems

This paper develops an exact formula for the fill rate of a single-stage inventory system that uses a general periodic-review base-stock policy. For normal demand, we present a fill-rate expression that uses the standard normal PDF and CDF, and develop two approximations for the fill rate.     

Optimal acquisition and production policy in a hybrid manufacturing/remanufacturing system with core acquisition at different quality levels

We study the acquisition and production planning problem for a hybrid manufacturing/remanufacturing system with core acquisition at two (high and low) quality conditions. We model the problem as a stochastic dynamic programming, derive the optimal dynamic acquisition pricing and production policy, and analyze the influences of system parameters on the acquisition prices and production quantities. The production cost differences among remanufacturing high- and low-quality cores and manufacturing new products are found to be critical for the optimal production and acquisition pricing policy: the acquisition price of high-quality cores is increasing in manufacturing and remanufacturing cost differences, while the acquisition price of low-quality cores is decreasing in the remanufacturing cost difference between high- and low-quality cores and increasing in manufacturing and remanufacturing cost differences; the optimal remanufacturing/manufacturing policy follows a base-on-stock pattern, which is characterized by some crucial parameters dependent on these cost differences.     

Note on the fill rate of single-stage general periodic review inventory systems

For base-stock policies, Zhang and Zhang [J. Zhang, J. Zhang, Fill rate of single-stage general periodic review inventory systems, Operations Research Letters 35 (2007) 503-509] derive the fill rate, defined as the long-run average fraction of demand satisfied immediately. We derive the same expression for the fill rate defined as the ratio of expected demand satisfied immediately to expected demand, and generalize to (R,Q) policies.     

Optimal policy and stability regions for the single product periodic review inventory problem,with stationary demands

In this paper we present rules concerning the optimal policy and stability regions for the single product periodic review inventory problem with stationary demands, over a finite horizon. The key parameter to the whole study is the Lot-Sizing Index (LSI) introduced by Blackburn and Millen. Two algorithms are presented. The first one constructs stability regions which are expressed as intervals of the LSI parameter, covering the whole range of its values. The proposed algorithm is very simple to understand and implement, and most importantly, it provides a solution table which can be used by the decision maker to easily determine the optimal policy for any problem with a given horizon and any possible combination of its cost parameters, namely any LSI value. The second proposed algorithm determines the optimal policy for any given LSI value; it constitutes a completely different approach to that of the Wagner–Whitin algorithm and requires very little computational effort.     

Optimal control policy for a standing order inventory system

In this paper, we consider a standing order inventory system in which an order of fixed size arrives in each period. Since demand is stochastic, such a system must allow for procurement of extra units in the case of an emergency and sell-offs of excess inventory. Assuming the average-cost criterion, Rosenshine and Obee (Operations Research 24 (1976) 1143–1155) first studied such a system and devised a 4-parameter inventory control policy that is not generally optimal. The current paper uses dynamic programming to determine the optimal control policy for a standing order system, which consists of only two operational parameters: the dispose-down-to level and order-up-to level. Either the average-cost or discounted-cost criterion can be assumed in the proposed model. Also, both the backlogged and lost-sales problems are investigated in this paper. By using a convergence theorem, we stop the dynamic programming computation and obtain the two optimal parameters.     

Optimal control of a finite-capacity inventory system with setup cost and lost sales

One of the most fundamental results in inventory theoryis the optimality of (s, S) policy for inventory systems withsetup cost. This result is established based on a key assumptionof infinite production/ordering capacity. Several studies haveshown that, when there is a finite production/ordering capacity,the optimal policy for the inventory system is very complicatedand indeed, only partial characterization for the optimal policyis possible. In this paper, we consider a continuous reviewinventory system with finite production/ordering capacity andsetup cost, and show that the optimal control policy for thissystem has a very simple structure. We also develop efficientalgorithms to compute the optimal control parameters.     

Periodic review lost-sales inventory models with compound Poisson demand and constant lead times of any length

In almost all literature on inventory models with lost sales and periodic reviews the lead time is assumed to be either an integer multiple of or less than the review period. In a lot of practical settings such restrictions are not satisfied. We develop new models allowing constant lead times of any length when demand is compound Poisson. Besides an optimal policy, we consider pure and restricted base-stock policies under new lead time and cost circumstances. Based on our numerical results we conclude that the latter policy, which imposes a restriction on the maximum order size, performs almost as well as the optimal policy. We also propose an approximation procedure to determine the base-stock levels for both policies with closed-form expressions.     

Cooperation and game-theoretic cost allocation in stochastic inventory models with continuous review

We study cooperation strategies for companies that continuously review their inventories and face Poisson demand. Our main goal is to analyze stable cost allocations of the joint costs. These are such that any group of companies has lower costs than the individual companies. If such allocations exist they provide an incentive for the companies to cooperate.     

Optimal ordering and pricing policy for an inventory system with trial periods

It is a business practice that home shopping companies offer a free trial period for their products with a goal of increasing sales. Under this policy, if for any reason customers are not satisfied with the purchase, they can return the product for a refund within the trial period. To develop inventory strategies in such environment, home shopping companies should take the return phenomenon into account so as to increase their profit. This paper considers this phenomenon and develops a seasonal inventory model to deal with the problem. Two scenarios are analyzed. In the first scenario, demand is assumed to be linearly price-dependent while in the second one, it is assumed to be exponentially price-dependent. The purpose of this research is to maximize the total profit over a given planning period by determining the optimal ordering quantity and price. The analytical results demonstrate that the optimal ordering quantity and prices are obtained using closed-form formulas.     

Effective and simple EOQ-like solutions for stochastic demand periodic review systems

In this paper we examine a periodic review system under stochastic demand with variable stockout costs. The optimal values for cycle length and amount of safety stock are difficult to obtain because one of the First Order Conditions does not have a closed form solution. However, by using a Taylor series expansion to approximate part of the cost function, we produce a simple cost function structure which is similar to that of deterministic models.We argue that this simple structure is also beneficial to promote the solution in other problems where coordination of cycles is required. To illustrate, we use the joint replenishment problem for multiple items under stochastic demand and suggest simple and efficient solution procedures.     

An order expediting policy for continuous review systems with manufacturing lead-time

Expedited shipments are often seen in practice. When the inventory level of an item gets dangerously low after an order has been placed, material managers are often willing to expedite the order at extra fixed and/or variable costs. This paper proposes a single-item continuous-review order expediting inventory policy, which can be considered as an extension of ordinary (s,Q)$(s\text{,}Q)$ models. Besides the two usual operational parameters: reorder point s and order quantity Q, it consists of a third parameter called the expedite-up-to level R. If inventory falls below R at the end of the manufacturing lead-time, the buyer can request the upstream supplier to deliver part of an outstanding order via a fast transportation mode. The amount expedited will raise inventory to R, while the remaining order is delivered via a slow (regular) supply mode. Simple procedures are developed to obtain optimal operational parameters. Computational results show that the proposed policy can save large costs for a firm if service level is high, demand variability is large, the extra cost for expediting is small, or the manufacturing lead-time is long.     

Cost-effective inventory control in a value-added manufacturing system

This paper considers the cost-effective inventory control of work-in-process (WIP) and finished products in a two-stage distributed manufacturing system. The first stage produces a common WIP, and the second stage consists of several production sites that produce differentiated products with different capacity and service level requirements. The unit inventory holding cost is higher at the second stage. This paper first uses a network of inventory-queue model to evaluate the inventory cost and service level achievable for given inventory control policy, and then derives a very simple algorithm to find the optimal inventory control policy that minimizes the overall inventory holding cost and satisfies the given service level requirements. Some managerial insights are obtained through numerical examples.     

A computational study on fill rate expressions for single-stage periodic review under normal demand and constant lead time

This paper reviews fill rate expressions for a single stage periodic review inventory system under normal demand and constant lead time, discusses the relationship among all expressions in the literature, and evaluates their robustness and accuracy. Monte Carlo simulation is used to numerically compare all expressions. We present conditions under which some expressions produce higher values than others.     

Optimal inventory policy with supply uncertainty and demand cancellation

We consider a periodic review model where the firm manages its inventory under supply uncertainty and demand cancellation. We show that because of supply uncertainty, the optimal inventory policy has the structure of re-order point type. That is, we order if the initial inventory falls below this re-order point, otherwise we do not order. This is in contrast to the work of Yuan and Cheung (2003) who prove the optimality of an order up to policy in the absence of supply uncertainty. We also investigate the impact of supply uncertainty and demand cancellation on the performance of the supply chain. Using our model, we are able to quantify the importance of reducing the variance of either the distribution of yield or the distribution of demand cancellation. The single, multiple periods and the infinite horizon models are studied.     

Heuristics with guaranteed performance bounds for a manufacturing system with product recovery

We consider a manufacturing system with product recovery. The system manufactures a new product as well as remanufactures the product from old, returned items. The items remanufactured with the returned products are as good as new and satisfy the same demand as the new item. The demand rate for the new item and the return rate for the old item are deterministic and constant. The relevant costs are the holding costs for the new item and the returned item, and the fixed setup costs for both manufacturing and remanufacturing. The objective is to determine the lot sizes and production schedule for manufacturing and remanufacturing so as to minimize the long-run average cost per unit time. We first develop a lower bound among all classes of policies for the problem. We then show that the optimal integer ratio policy for the problem obtains a solution whose cost is at most 1.5% more than the lower bound.     

On sufficient conditions for cost rate function unimodality in periodic review (s, S) inventory models

The cost rate function that arises in the stationary analysis of a class of periodic review regenerative inventory systems is known to be unimodal if the renewal density of the underlying demand sequence is decreasing. We prove that the same result holds, under zero leadtimes, if the renewal density is concave increasing.     

Fuzzy periodic review system with fuzzy random variable demand

In this paper, a periodic review inventory system has been analyzed in a mixed imprecise and uncertain environment where fuzziness and randomness appear simultaneously. A model has been developed with customer demand assumed to be a fuzzy random variable. The lead-time has been assumed to be a constant. The lead-time demand and the lead-time plus one period’s demand have also been assumed to be fuzzy random variables. A methodology has been developed to determine the optimal inventory level and the optimal period of review such that the total expected annual cost in the fuzzy sense is minimized. A numerical example has been presented to illustrate the model.