Optimal two-stage lot sizing and inventory batching policies

This paper extends previous studies of two-stage lot sizing problems with finite production rates. We develop various lot sizing and inventory batching (i.e., operation–unit batching (OUB) and unit–unit batching (UUB)) models under different system characteristics and lot sizing and inventory policies. The analysis of the optimality of the lot size ratio between the two stages reveals (1) that both non-increasing and non-decreasing lot sizing policies can be optimal in both OUB and UUB, (2) that a non-integer lot size ratio can be optimal in OUB, and (3) that an integer lot size ratio is always optimal in UUB. We present a simple, easy-to-implement, optimal solution approach to the two-stage lot sizing and inventory batching problem, along with examples.     

Optimal inventory and pricing policies for remanufacturable leased products

In this paper we consider a company which leases new products and also sells remanufactured versions of the new product that become available at the end of their lease periods. When the amount of end-of-lease items in stock is not sufficient to meet the demand for remanufactured products, the firm may purchase additional cores from a third-party supplier. We develop a dynamic programming formulation for determining the optimal price of remanufactured products, and optimal payment structure for the leased products. Our objective is to maximize the discounted system-wide profit over a finite horizon. The profit function consists of revenues that are obtained from remanufactured product sales and leasing, remanufacturing and manufacturing costs, inventory holding and shortage costs. We consider a consumer choice based demand model for mapping a potential customer into one of the product segments (a remanufactured product customer or a customer for a leased product with a particular lease period) for a given price/lease payment vector. We explore several properties of the discounted profit function and provide insight on the behavior of pricing and inventory policies. We also investigate the effect of key product characteristics such as deterioration in age, cost of shortage in remanufacturable product inventory, and key market characteristics such as relative willingness-to-pay for buying a remanufactured product and relative willingness-to-pay for leasing a new product on optimal pricing policies through a computational study.     

Base-stock policies for the lost sales inventory system with Poisson demand and Erlangian lead times

The base-stock policies for the studied inventory system can easily be evaluated through Erlang's loss formula when the lead times are mutually independent. This is often the case only if the base-stock $S$ is one. If $S$ is larger than one, the Erlangian lead times become stochastically dependent under the realistic assumption that the replenishment orders do not cross in time. We make this assumption and show for any positive $S$ that the number of replenishment orders outstanding has an equilibrium distribution which is a slightly modified truncated version of a negative binomial distribution. It turns out to be easy to compute the stock-out frequency recursively for $S=1,2,\dots .$ For each $S$, the average stock can be specified in terms of this frequency. We prove that the frequency is convex in $S$. It is therefore straightforward to compute the base-stock for which the average cost is minimized and to compute the minimum average cost. Our numerical study illustrates that the minimum average cost is very sensitive to the shape parameter describing the Erlangian lead times, which is in sharp contrast to the complete insensitivity when lead times are independent.     

Trend breaks in US inventory to sales ratios

This paper documents econometrically that US Inventory to Sales Ratios have time trends and that there have been significant breaks in these time trends at both the aggregate and industry level. Using techniques developed by Bai (Rev. Econ. Stat. (1997) 551) and by Bai and Perron (Econometrica 66 (1) (1998) 47), we simultaneously estimate the time trend parameters and the dates of statistically significant breaks in those trends. Using data from 1967 through 2001, it is estimated that aggregate Manufacturing and Trade Inventory to Sales ratio trended upwards at a 0.76%/year rate before 1985 Q3 and has trended downwards at a 1.29%/year rate since then. Downtrends are estimated to have begun in Manufacturing ratios in 1983, Wholesalers in 1985, and Retailers in 1990. Before the mid-1990s nearly all the reductions in ratios occurred in industries carrying durable goods. Examination of the timing and possible explanations of trend breaks for the 20 two-digit manufacturing industries suggests that breaks are much more likely to have occurred in recessions and may be related to changes in the variance of the industry's sales growth rate.     

Integer-ratio policies for distribution/inventory systems

We study a multistage distribution/inventory system with a central warehouse and N retailers. Customer demand arrives at each retailer at a constant rate. The retailers replenish their inventories from the warehouse, which in turn orders from an outside supplier. It is assumed that shortages are not allowed and lead times are negligible. The goal is to determine policies which minimize the overall cost in the system, that is, the sum of the costs at each facility consisting of a fixed charge per order and a holding unit cost. We propose a heuristic procedure to compute near-optimal policies. Computational results on several randomly generated problems are reported.     

Joint determination of inventory replenishment and sales effort with uncertain market responses

An optimal joint operational and marketing decision is crucial for robust supply chain management. This paper addresses concurrent determination of inventory replenishment and sales effort decisions such as price, incentives to salesforce, and short-term promotions, or a combination of them. Market responses to sales efforts are typically highly uncertain, and demand in each period has its distribution dependent on the selected sales effort. In each period a replenishment order may be issued, which incurs both fixed and variable ordering costs, and at the same time the sales effort is also determined, the execution of which may incur costs. For such a model, the previously developed methods which are used for the joint inventory-pricing models become inadequate. A computational procedure for obtaining an optimal joint policy is addressed, and the conditions for the optimality of that policy are identified.     

Optimal inventory policy with noninstantaneous receipt under trade credit

In the last two decades, the models for inventory replenishment policies under trade credit have been widely studied by several researchers. However, the extant papers only consider the effects of permissible delay in payments in the inventory systems. In some situations, the supplier also may offer a cash discount to encourage retailer to pay for his purchases quickly. On the other hand, an inventory model with instantaneous receipt is not a common feature in actual practice. For generality, this study develops an inventory model with noninstantaneous receipt under trade credit, in which the supplier provides not only a permissible delay but also a cash discount to the retailer. We then characterize the optimal solution and provide an easy-to-use criterion to find the optimal order strategies. Finally, several numerical examples are presented to illustrate the theoretical results.     

Near cost-optimal inventory control policies for divergent networks under fill rate constraints

We deal with the optimisation of stock levels in general divergent networks under a periodic review, order-up-to ($\text{R,}\text{S}$) policy. The goal is to attain target fill rates, while the total holding costs in the entire network are minimised. To this end, we first present a method for the fast calculation of the control parameters, given central and intermediate stock levels. Next we develop an approximate procedure to determine stock levels sequentially. Extensive numerical experimentation shows that this procedure yields satisfactory results. It also shows that significant stocks at intermediate stockpoints are only useful if unit holding costs in these stockpoints are considerably less than in the end stockpoints that deliver directly to the final customers.     

Robust optimal policies of production and inventory with uncertain returns and demand

We consider an inventory and production planning problem with uncertain demand and returns, in which the product return process is integrated into the manufacturing process over a finite planning horizon. We first propose an inventory control model for the return and remanufacturing processes with consideration of the uncertainty of the demand and returns. Then a robust optimization approach is applied to deal with the uncertainty of the problem through formulating a robust linear programming model. Moreover, properties on the robust optimization model are studied, and an equivalent robust optimization model based on duality theory is obtained which allows the solutions to be derived more efficiently. Finally, we provide a set of numerical examples to verify the effectiveness of the approach and analyze the effects of the key parameters on the solutions.     

Optimal inventory control of empty containers in inland transportation system

In this paper, we deal with an inventory control problem of empty containers in an inland transportation system. In inland container transportation, freights (containers) are transported between terminal and the customer’s location by trucks, trains and barges. Empty containers are an important logistic resource and shipping companies try to operate and manage empty containers efficiently. Because of the trade imbalance between hub ports, empty containers should be periodically repositioned from surplus areas to shortage areas. However, it is not easy to exactly forecast the demand of empty containers, and we therefore need to build an efficient way to reposition the empty containers. In this paper, we consider a shortage area and propose an efficient inventory policy to control empty containers. We assume that demands per unit time are independent and identically distributed random variables. To satisfy the demand of empty containers, we reposition empty containers from other hubs based on the (s, S) inventory policy, and also consider the lease of empty containers with zero lead time. For the leased containers, we should return the number of empty containers leased to the leaser after the specified period. For a given policy, simulation is used to estimate the expected cost rate and we use the optimization tool, OptQuest^® in Arena to obtain the near optimal (s, S) policy in numerical examples.     

Batch quantities when forecasts are improving

This paper considers the choice of initial batch quantities when demand forecasts are improving. We assume that the standard deviation of the demand per period is decreasing exponentially and approaching a long-run value. A discrete time stochastic single-level inventory model is considered. There are traditional holding and backorder costs as well as an ordering cost. The ordering periods must be determined in advance, and we wish to determine a suitable schedule.     

Optimal ordering policies in response to a discount offer

Sometimes supplier offers a temporary price discount to increase cash flow or decrease the inventory level of certain items. Thus, the manufacturer may be able to improve the effectiveness of his inventory system by ordering a special quantity at this sale period. In this paper, economic order quantity (EOQ) models with a discounted price are developed to obtain the optimal ordering policy during the sale period for five different cases: (a) coincidence of sale period with replenishment time, (b) non-coincidence of sale period with replenishment time, (c) sale period is longer than a cycle, (d) discounted price as a function of the special ordering quantity, and (e) incremental discount. Each case has its own characteristics of the sale period and the discounted price. The objective is to take the maximum possible advantage from the discounted price by ordering a special quantity during the sale period. The optimal ordering policy is obtained by maximizing the difference between the two costs: Regular EOQ cost and special quantity cost during the sale period. Moreover, a comparison of different discount scenarios is developed to sense the effect of different parameters on the ordering policies. The annual gain obtained is linearly related to the discount and the on-hand remnant inventory. Numerical analyses are provided to illustrate and testify the values of the optimal special quantity. The analysis showed an impressive improvement in the effectiveness of the inventory system when a special order is placed during the sale period. The optimal special quantity is driven for each case to visualize real-life problems. Sensitivity analysis is also initiated to study the change in the total savings with respect to the variation of the special optimal quantity.     

Price competition when three are few and four are many

I examine price competition in a market for a homogeneous good when consumers observe prices subject to a random shock (perception error). When firms have symmetric costs, there exists a unique equilibrium in pure strategies, which is symmetric. When there are up to three sellers in the market, the sellers extract the entire consumer surplus. However, with at least four firms, assuming that the marginal cost is sufficiently low relative to consumers’ valuation, both consumers and producers may enjoy a positive surplus. The marginal-cost pricing is never observed in an equilibrium with finitely many firms. Potential policy implications are discussed.     

German inventory to sales ratios 1971-2005—An empirical analysis of business practice

The purpose of this study is to test empirically for the first time the general hypothesis that inventory to sales ratios have decreased over time in the German economy. Although inventory reduction has been a prevalent topic in the production and operations management literature, there is a lack of empirically confirmed answers to questions. They are as follows: Have inventories in German firms decreased overall during the past decades? What sectors of German industry are leading (lagging behind) inventory reduction? Has inventory reduction developed differently for raw materials, work-in-process, or finished goods? In which periods was marginal inventory reduction greatest? To the best of our knowledge, this empirical study is the first to broadly investigate inventory development from the 1970s until the present for a major European economy, Germany, and will provide the first answers to the research questions stated above using aggregate industry-level data provided by the Deutsche Bundesbank. We show that inventory levels decreased overall in many sectors of German industry. This reduction was mainly marked for raw materials and finished goods, particularly for the second-half of the time frame investigated.     

Exact and approximate calculation of the cycle service level in periodic review inventory policies

The parameters of stock policies are usually determined to minimize costs while satisfying a target service level. In a periodic review policy the time between reviews can be selected to minimize costs while the order-up-to-level is based on the fulfilment of a target service level. Generally, the calculation of this service measurement is obtained using approximations based on an additional hypothesis related to the demand pattern. Previous research has shown that there is a substantial difference between exact and approximate calculations in some general circumstances, so in these cases the service level is not accomplished or the stock level is overestimated. Although an exact calculation of CSL was developed in previous work, the computational effort required to apply it in practical environments leads to the proposal of two approximate methods (PI and PII) that, with the classic approximation, are analysed and evaluated in this paper. This analysis points out the risks of using the classic approximation and leads one to suggest PII as the most suitable and accurate enough procedure to compute the CSL straightforwardly in practice. Additionally, a heuristic approach based on PII is proposed to accept or reject an inventory policy in terms of fulfilling a given target CSL. This paper focuses on uncorrelated, discrete and stationary demand with a known distribution pattern and without backlog.     

Quality Leadership when Regulatory Standards are Forthcoming

In many markets, governments set minimum quality standards while some sellers compete on the basis of quality by exceeding them. Such quality leadership strategies often win public acclaim, especially when they involve environmental attributes. Using a duopoly model of vertical product differentiation, we show that if the high-quality firm can commit to a quality level before regulations are promulgated, it induces the regulator to weaken standards, and welfare falls. Our results raise doubts about the social benefits of corporate self-regulation, and highlight the dangers of lengthy delays between legislative mandates for new regulations and their implementation.     

Pricing network elements when costs are changing

This paper compares the pricing rule embodied in the Federal Communications Commissions (FCCs) forward-looking cost model with a competitive equilibrium pricing rule. The rules differ due to differences in discount factors, the time path of operating cost recoveries, and the methods used to spread capital costs over time. A calibrated comparison of the rules for end-office switching reveals that the latter is most significant, causing the FCCs rule to understate cost recoveries by 24 percent. A rough aggregation suggests this difference compounds to billions of dollars nationwide, solely for the direct costs of end-office switching. This understatement is driven by falling switch costs over time, and the FCCs rule would overstate cost recoveries by a similar magnitude for assets whose costs rise over time. The competitive equilibrium rule also clarifies the endogeneity of economic life and reveals that depreciation assumptions are not needed for calculating competitive equilibrium prices.     

Optimal policies in hybrid manufacturing/remanufacturing systems with product substitution

In hybrid control systems for simultaneous remanufacturing of used products and manufacturing of new ones, the two operations are not directly interconnected if remanufactured items are downgraded and have to be sold in markets different from those for new products. Sometimes a connection between these markets is given by a downward substitution property which allows the producer to offer a new item instead of a remanufactured one in case of a shortage of a remanufactured product. Thus, shortage costs can be avoided, but a loss in profit due to sale of a high-graded product at the price of a low-graded one has to be accepted. For a single-period problem with stochastic returns of used products and stochastic demands of serviceable ones, it is shown how the manufacturing and remanufacturing decisions have been coordinated in order to maximize the total expected profit. It turns out that under strictly proportional costs and revenues a medium-simple ‘order-up-to policy’ with two parameters and two parameter functions is optimal. However, optimal policies in situations where manufacturing leadtimes exceed leadtimes for remanufacturing turn out to be different from those in the opposite leadtime case. The research presented combines methods for policy analysis in stochastic manufacturing/remanufacturing problems and in stochastic inventory control problems with substitutable products.     

Airport congestion and capacity when carriers are asymmetric

This study investigates pricing and capacity investment for a congested airport served by multiple carriers. Removing the symmetric carrier assumption, when airlines’ market shares are potentially asymmetric, the socially non-discriminatory optimal charge rule should include an additional term that corrects the distortion caused by pricing itself. The first-best outcome cannot be achieved, and the airport overinvests under the non-discriminatory optimal charge. However, if the airport levies discriminatory charges respective to carriers, the first-best outcome can be achieved and capacity investment is socially efficient under discriminatory optimal charges. In addition, the discriminatory optimal charges levied on a carrier with a larger (smaller) market share are lower (higher).