On the supplier diversification under binomial yield

In a recent article, Fadiloglu et al. study an EOQ inventory model with multiple suppliers and binomial yields and conclude that sole sourcing is always optimal. This finding is contrary to the literature advocating diversification of the supply base in the presence of uncertainty. In this note, we present an alternative and more elegant proof of the optimality of sole sourcing in the EOQ model with binomial yield. Our approach extends this result to a general structure for the fixed order cost and provides more insights into the problem.     

A note on supplier diversification under random yield

Fad?lo?lu et al. and Tajbakhsh et al. consider supplier diversification in an EOQ type inventory setting with multiple suppliers and binomial yield and show that working with a single supplier is always optimal. In this short note, we present an alternative and more elegant proof of the optimality of sole sourcing in the EOQ model with general random yield.     

Concepts for safety stock determination under stochastic demand and different types of random production yield

We consider a manufacturer’s stochastic production/inventory problem under periodic review and present methods for safety stock determination to cope with uncertainties that are caused by stochastic demand and different types of yield randomness. Following well-proven inventory control concepts for this problem type, we focus on a critical stock policy with a linear order release rule. A central parameter of this type of policy is given by the safety stock value. When non-zero manufacturing lead times are taken into account in the random yield context, it turns out that safety stocks have to be determined that vary from period to period. We present a simple approach for calculating these dynamic safety stocks for different yield models. Additionally, we suggest approaches for determining appropriate static safety stocks that are easier to apply in practice. In a simulation study we investigate the performance of the proposed safety stock variants.     

Optimal control of an inventory-production system with state-dependent random yield

In this paper we study a single stage, periodic-review inventory problem for a single item with stochastic demand. The inventory manager determines order sizes according to an order-up-to logic and observes a random yield due to quality problems in the production. We distinguish between two different states of the production process combined with different probabilities to produce a defective unit. In order to improve the production process, periodic inspections are conducted and in case of a failure the machine is repaired. Approximations are developed to evaluate the average cost for a given order-up-to level and a given inspection interval and we illustrate the existence of optimal policy parameters. The approximations are tested in a simulation study and reveal an excellent performance as they lead to near optimal policy parameters. Moreover, we decompose the problem and test different methods to compute the policy parameters either sequentially or separately. Our results show that a joint optimization of the inventory and maintenance policy leads to a better system performance and reduced costs.     

Lot sizing with random yield and different qualities

This paper considers a production/inventory system where items produced/purchased are of different qualities: Types A and B. Type A items are of perfect quality, and Type B items are of imperfect quality; but not necessarily defective; and have a lower selling price. The percentage of Type A (the yield rate) is assumed to be a random variable with known probability distribution. The electronics industry gives good examples of such situations. We extend the classical single period (newsvendor) and the economic order quantity (EOQ) models by accounting for random supply and for imperfect quality (Type B) items which are assumed to have their own demand and cost structure. We develop mathematical models and prove concavity of the expected profit function for both situations. We also present detailed analysis and numerical results. We focus on comparing the profitability of the novel proposed models with models from the literature (and derivatives of these models) that develop the optimal order quantity based on the properties of Type A items only (and ignore Type B items). We find that accounting for Type B items can significantly improve profitability.     

Optimal policies for inventory systems with discretionary sales, random yield and lost sales

We determine replenishment and sales decisions jointly for an inventory system with random demand, lost sales and random yield. Demands in consecutive periods are independent random variables and their distributions are known. We incorporate discretionary sales, when inventory may be set aside to satisfy future demand even if some present demand may be lost. Our objective is to minimize the total discounted cost over the problem horizon by choosing an optimal replenishment and discretionary sales policy. We obtain the structure of the optimal replenishment and discretionary sales policy and show that the optimal policy for finite horizon problem converges to that of the infinite horizon problem. Moreover, we compare the optimal policy under random yield with that under certain yield, and show that the optimal order quantity （sales quantity） under random yield is more （less） than that under certain yield.     

An optimal replenishment policy for an EOQ model with partial backlogging

We study an inventory system where demand on the stockout period is partially backlogged. The backlogged demand ratio is a mixture of two exponential functions. The shortage cost has two significant costs: the unit backorder cost (which includes a fixed cost and a cost proportional to the length of time for which the backorder exists) and the cost of lost sales. A general procedure to determine the optimal policy and the minimum inventory cost for all the parameter values is developed. This model generalizes several inventory systems analyzed by different authors. Numerical examples are used to illustrate the theoretical results.     

Modeling and analysis for determining optimal suppliers under stochastic lead times

The policy of simultaneously splitting replenishment orders among several suppliers has received considerable attention in the last few years and continues to attract the attention of researchers. In this paper, we develop a mathematical model which considers multiple-supplier single-item inventory systems. The item acquisition lead times of suppliers are random variables. Backorder is allowed and shortage cost is charged based on not only per unit in shortage but also per time unit. Continuous review (s,Q)$(s\text{,}Q)$ policy has been assumed. When the inventory level depletes to a reorder level, the total order is split among n suppliers. Since the suppliers have different characteristics, the quantity ordered to different suppliers may be different. The problem is to determine the reorder level and quantity ordered to each supplier so that the expected total cost per time unit, including ordering cost, procurement cost, inventory holding cost, and shortage cost, is minimized. We also conduct extensive numerical experiments to show the advantages of our model compared with the models in the literature. According to our extensive experiments, the model developed in this paper is the best model in the literature which considers order splitting for n-supplier inventory systems since it is the nearest model to the real inventory system.     

Inventory returns and special sales in a lot-size system with constant rate of deterioration

The problem of returning or of selling the inventory excess to optimal stock level is considered for deteriorating items. Two inventory models, viz. the infinite and the finite horizon models, are developed, in which the deterioration is assumed to be a constant fraction of the on hand inventory. Both the models are developed under the assumptions of instantaneous delivery and no shortages. When there is no deterioration, the developed models are related to the corresponding inventory models for non-deteriorating items. Examples are given to illustrate the derived results.     

Economic order quantity under conditionally permissible delay in payments

Within the economic order quantity (EOQ) framework, the main purpose of this paper is to investigate the retailer’s optimal replenishment policy under permissible delay in payments. All previously published articles dealing with optimal order quantity with permissible delay in payments assumed that the supplier only offers the retailer fully permissible delay in payments if the retailer ordered a sufficient quantity. Otherwise, permissible delay in payments would not be permitted. However, in this paper, we want to extend this extreme case by assuming that the supplier would offer the retailer partially permissible delay in payments when the order quantity is smaller than a predetermined quantity. Under this condition, we model the retailer’s inventory system as a cost minimization problem to determine the retailer’s optimal inventory cycle time and optimal order quantity. Three theorems are established to describe the optimal replenishment policy for the retailer. Some previously published results of other researchers can be deduced as special cases. Finally, numerical examples are given to illustrate all these theorems and to draw managerial insights.     

The deterministic EOQ with partial backordering: A new approach

Previous authors have shown that if demand that cannot be filled from stock is partially backordered, then using the full-backordering model or assuming that all stockouts will result in lost sales can lead to substantial increases in cost relative to using a model that specifically recognizes the percentage of the stockouts that will be backordered. The models that these authors developed resulted in procedures or equations that are relatively difficult to use. In this paper we take a different approach to modeling the deterministic EOQ with partial backordering that results in equations that are more like the comparable equations for the basic EOQ and its full-backordering extension.     

On coordinating an assembly system under random yield and random demand

We consider a decentralized assembly system in which the customer demand and the yield of the suppliers are random. We establish the concavity of expected supply chain profit for arbitrary number of suppliers. We propose two contracts and show that they coordinate the chain under forced compliance. The contracts are mixed type of contracts that include payments from different contract schemes. Particularly, a payment or a penalty to the worst performing supplier seems inevitable. Apart from providing a coordinating contract, we also provide qualitative insights based on a numerical illustration of centralized and decentralized solutions.     

A new solution method for the finite-horizon discrete-time EOQ problem

We consider the finite-horizon discrete-time economic order quantity problem. Kovalev and Ng (2008) have developed a solution approach for solving this problem. Their approach requires a search for the optimal number of orders, which takes O(logn)$\text{O}(\log n)$ time. In this note, we present a modified solution method, which can determine the optimal solution without the need of such a search.     

A note on coordination in decentralized assembly systems with uncertain component yields

Gurnani and Gerchak [H. Gurnani, Y. Gerchak, Coordination in decentralized assembly systems with uncertain component yields, European Journal of Operational Research 176 (2007) 1559–1576] study coordination of a decentralized assembly system in which the demand of the assembler is deterministic and the component yields are random. They present incentive alignment control mechanisms under which system coordination is achieved. In this note, we extend Gurnani and Gerchak’s model to the case of positive salvage value and n asymmetric suppliers, and show that the shortage penalty contract which can coordinate Gurnani and Gerchak’s model no longer coordinates the extended model. Furthermore, we present a new kind of contract, surplus subsidy contract, to coordinate the extended model and prove that the profit of the supply chain under coordination can be arbitrarily divided between the component suppliers and the assembler.     

On an EPQ model for deteriorating items under permissible delay in payments

This paper derives a production model for the lot-size inventory system with finite production rate, taking into consideration the effect of decay and the condition of permissible delay in payments, in which the restrictive assumption of a permissible delay is relaxed to that at the end of the credit period, the retailer will make a partial payment on total purchasing cost to the supplier and pay off the remaining balance by loan from the bank. At first, this paper shows that there exists a unique optimal cycle time to minimize the total variable cost per unit time. Then, a theorem is developed to determine the optimal ordering policies and bounds for the optimal cycle time are provided to develop an algorithm. Numerical examples reveal that our optimization procedure is very accurate and rapid. Finally, it is shown that the model developed by Huang [1] can be treated as a special case of this paper.     

The inventory models under conditional trade credit in a supply chain system

In several recent investigations dealing with the economic order quantity with permissible delay in payments, the following assumptions are made:     

Confidence-based optimisation for the newsvendor problem under binomial,Poisson and exponential demand

We introduce a novel strategy to address the issue of demand estimation in single-item single-period stochastic inventory optimisation problems. Our strategy analytically combines confidence interval analysis and inventory optimisation. We assume that the decision maker is given a set of past demand samples and we employ confidence interval analysis in order to identify a range of candidate order quantities that, with prescribed confidence probability, includes the real optimal order quantity for the underlying stochastic demand process with unknown stationary parameter(s). In addition, for each candidate order quantity that is identified, our approach produces an upper and a lower bound for the associated cost. We apply this approach to three demand distributions in the exponential family: binomial, Poisson, and exponential. For two of these distributions we also discuss the extension to the case of unobserved lost sales. Numerical examples are presented in which we show how our approach complements existing frequentist—e.g. based on maximum likelihood estimators—or Bayesian strategies.     

Random yield supply chain with a yield dependent secondary market

This paper considers a simple supply chain with one supplier and one retailer where the supplier’s production is subject to random yield and the retailer faces uncertain demand. There exists a secondary market for acquiring or disposing products by the supplier. We study both the centralized and decentralized systems. In the decentralized system, a no risk sharing contract and a risk sharing minimum commitment contract are analyzed. The supply chain with the risk sharing contract is further analyzed with a constant secondary market price and a yield dependent secondary market price. We present both the supplier’s and the retailer’s optimal strategies and provide insights for managers when making decisions under random yield risk and demand uncertainty. We find that the secondary market generally has a positive impact on supply chain performance and the actual effect of random yield risk on the supply chain performance depends on cost parameters and supply chain contract settings. Under certain conditions, reducing yield randomness may weaken the double marginalization effect and improve the chain performance. From the numerical study, we also show that there exists an optimal commitment level for the supply chain.     

A note on optimal ordering policies when the supplier provides a progressive interest scheme

Goyal et al. [Goyal, S.K., Teng, J.T., Chang, C.T., 2007. Optimal ordering policies when the supplier provides a progressive interest scheme. European Journal of Operational Research 179, 404–413] explore optimal ordering policies when the supplier provides a progressive interest scheme. The main purpose of this paper is fourfold:

(1)

This paper simplifies the total relevant cost per year Z(T) of Goyal et al. (2007) such that we can locate the optimal solutions of Z(T) by an easier way.     

Impact of uncertainty in the quality of returns on the profitability of a single-period refurbishing operation

This paper investigates how the profitability of reuse activities is affected by uncertainty regarding the quality of returned products. Specifically, we examine a reverse supply chain consisting of two collection sites and a refurbishing site, which faces stochastic demand for refurbished products in a single-period setting. The quality of returns (refurbishing yield) becomes known only after the transportation of the products to the refurbishing site. We prove that the expected profit function has a unique optimal solution (procurement and production quantities) and we derive the conditions under which it is optimal to use only one of the collection sites. The analysis is supported by numerical results which provide insights regarding the effect of the uncertain yields at the two collection sites and their correlation on optimal decisions and system profitability.