Procurement strategies for lost-sales inventory systems with all-units discounts

Despite the prevalence of all-units discounts in procurement contracts, these discounts pose a technical challenge to analyze procurement strategies due to neither concave nor convex ordering costs. In this paper, we consider the optimal procurement strategies with all-units discounts under the lost-sales setting. By assuming log-concave demands, we find that the optimal procurement strategies have a generalized Q-jump (s, S) structure by introducing a new notion of Q-jump single-crossing. In particular, a sufficient condition is provided for degenerating the optimal procurement strategies from a generalized Q-jump (s, S) structure into a Q-jump (s, S) structure, which is definitely optimal for the single-period problem. Extensive numerical results suggest that the Q-jump (s, S) policy as a heuristic performs considerably well when its optimality sufficient condition is violated. Our results can be extended to systems with multi-break all-units discounts, and systems with all-units discounts on batch ordering.     

Improving service rate for a tree-type three-echelon supply chain system with backorders at retailer's level

This research deals with a distributive or tree-type three-echelon production-distribution supply chain system with allowable backorder. Allowing backorder could reduce the total of a production-distribution system by reducing holding cost due to the lower average inventory, even though backorders carry some cost and lower a company's goodwill. The main purpose of this research is to develop replenishment policies for a tree-type three-echelon supply chain system with allowable backorder. The supply chain network is composed of a producer, multiple distributors, and multiple retailers. This research attempts to improve service rate by reducing the backorder at the retailer level. The distributors are allowed to ship product to retailer quicker in order to reduce backorder. The total cost function of the proposed model is developed. Since the total cost function contains some integer variables, differentiating the function with respect to the variables could not be used as a basis to solve for the optimal solutions. A branching search process was utilized to find the integer solutions. A numerical example is used to demonstrate improvement in service rate and total cost using the model.     

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.     

An economic order quantity model with partial backlogging under general backorder cost function

In this work, we study a single-item inventory model where shortages are allowed. A known constant fraction of the demand during the stockout period is backlogged, and the rest are lost sales. Usually, in the literature on inventory control, the unit backorder cost is considered to be a linear function of the waiting time until the customer gets the item. However, in some real-world situations, the unit cost of a backorder may not be linear. To model this situation, we develop a new approach by considering that the backlogging unit cost is a nondecreasing, continuous, and positive function of the amount of time the customers wait before receiving the item. Our objective is to maximize the average profit per unit time. An effective solution procedure to determine the optimal policy and the maximum average profit is developed. Numerical examples, which help us to understand the theoretical results, are also presented.     

An economic order quantity with imperfect quality and quantity discounts

Previous studies in the issue of inventory models with imperfect quality assumed the defectives could be sold in a batch by the end of the inspection process and the manufacturing systems were push systems. However, the above assumptions may not be true in the pull system in which buyer is powerful. Therefore, in this paper, we develop a new inventory model for items with imperfect quality and quantity discounts where buyer has exerted power over its supplier. Based on the concept of powerful buyer, there are three considerations included in this new model: (1) the order quantity is manufactured at one setup and is shipped over multiple deliveries, (2) the defectives are screened out by a 100% inspection for each shipment but sold in a batch by the end of inspection at the last shipment of each cycle, and (3) the supplier offers quantity discounts to response the request of the powerful buyer. Further, an algorithm is developed to help the powerful buyer to determine the optimal order policy accurately and quickly. Two numerical examples are available in this paper to illustrate the proposed model and algorithm. Besides, based on the numerical examples, a sensitivity analysis is made to investigate the effects of four important parameters (the inspection rate, the defective rate, the receiving cost, and the ordering cost) on the optimal solution.     

A stochastic periodic review integrated inventory model involving defective items, backorder price discount, and variable lead time

The purpose of this article is to investigate a stochastic integrated supplier-retailer inventory problem. The model analyzed in this article explores the problem of the protection interval, the backorder price discount, the lead time, and the numbers of shipments from the supplier to the retailer in one production run as control variables to widen applications for an integrated periodic review inventory model. We consider the situation in which the supplier and the retailer establish a long-term strategic partnership and contract to jointly determine the best strategy. We assume that the protection interval demand follows a normal distribution. Our objective is to determine the optimal review period, the optimal backorder price discount, the optimal lead time, and the optimal number of shipments from the supplier to the retailer in one production run, so that the joint expected annual total cost incurred has the minimum value. Furthermore, an algorithm of finding the optimal solution is developed. Also, the sensitivity analysis included and a numerical example is given to illustrate the results of the proposed model.     

Nonlinear Programming Analysis to Estimate Implicit Inventory Backorder Costs

In this paper, we use nonlinear programming to provide an alternative treatment of the economic order quantity problem with planned backorders. Many businesses, such as capital-goods firms that deal with expensive products and some service industries that cannot store their services, operate with substantial backlogs. In practical problems, it is usually very difficult to estimate accurately the values of the two types of backorder costs, i.e., the time-dependent unit backorder cost and the unit backorder cost. We redefine the original problem without including these backorder costs and construct a nonlinear programming problem with two service measure constraints which may be easier to specify than the backorder costs. We find that, with this different formulation of our new problem, we obtain results which give implicit estimates of the backorder costs. The alternative formulation provides an easier-to-use model and managerially meaningful results. Next, we show that, for a wide range of parameter values, it usually suffices to consider only one type of backorder cost, or equivalently, only one type of service measure constraint. Finally, we develop expressions which bracket the optimal values of the decision variables in a narrow range and provide a simple method for computing the optimal solution. In the most complicated case, this method requires finding the unique root of a polynomial.     

马尔科夫决策过程在R/M系统库存控制中的应用

给出了有限阶段R/M(再制造/制造)系统集成库存模式,利用马尔科夫决策理论对库存容量有限、具有随机再制造零部件的R/M系统集成库存进行研究,考虑再制造零部件单位成本、新零部件可变费用和固定费用、超过库存容量的那部分再制造零部件的附加费用及有缺货赔偿的情况,给出了马尔科夫决策过程模型,得到了随机最优控制策略.     

Some general properties for the newsboy problem with an extraordinary order

In this paper, an extension of the standard newsboy problem is presented, involving an extraordinary order and a variable mixture of backorders and lost sales. The backlogged demand ratio is given by a nonincreasing function of the quantity of shortage. Some general properties for the expected cost are derived under weak assumptions about the backorder rate function. When the backorder rate is a linear function, some sufficient conditions for the global convexity of the expected cost are derived. A sufficient condition for the local concavity of this function is also provided. Numerical examples are presented to illustrate the theoretical results and a specific practical case is proposed and solved. Moreover, a sensitivity analysis of the optimal solution with respect to the parameters of the backorder rate function is included. Finally, some extensions of the proposed model are suggested as possible directions for future research.     

Solving the capacitated lot-sizing problem with backorder consideration

In this research, we formulate and solve a type of the capacitated lot-sizing problem. We present a general model for the lot-sizing problem with backorder options, that can take into consideration various types of production capacities such as regular time, overtime and subcontracting. The objective is to determine lot sizes that will minimize the sum of setup costs, holding cost, backorder cost, regular time production costs, and overtime production costs, subject to resource constraints. Most existing formulations for the problem consider the special case of the problem where a single source of production capacity is considered. However, allowing for the use of alternate capacities such as overtime is quite common in many manufacturing settings. Hence, we provide a formulation that includes consideration of multiple sources of production capacity. We develop a heuristic based on the special structure of fixed charge transportation problem. The performance of our algorithm is evaluated by comparing the heuristic solution value to lower bound value. Extensive computational results are presented.     

A comprehensive note on: An economic order quantity with imperfect quality and quantity discounts

Lin [T.Y. Lin, An economic order quantity with imperfect quality and quantity discounts, Appl. Math. Model. 34 (10) (2010) 3158–3165] recently proposed an EOQ model with imperfect quality and quantity discounts, where the lot-splitting shipments policy is adopted. In this note we first rectify the holding cost terms showed in Lin to obtain a new objective function, then resolve the problem and develop an easy to implement algorithm to find the overall optimal solutions for the model. Besides, we present a new model for items with imperfect quality, where lot-splitting shipments and different holding costs for good and defective items are considered. The closed-form formulas for determining the optimal ordering and shipping policies are derived. Also, the results are examined analytically and numerically to gain more insights of the solutions.     

(<Emphasis Type="Italic">Q,r</Emphasis>) Inventory Model with a Mixture of Lost Sales and Time-Weighted Backorders

This paper presents a stochastic inventory model for situations in which, during a stockout period, a fraction β of the demand is backordered and the remaining fraction 1 – β is lost. The model is suggested by the customers' different reactions to a stockout condition: during the stockout period, some patient customers wait until their demand is satisfied, while other impatient or urgent customers cannot wait and have to fill their demand from another source. The cost of a backorder is assumed to be proportional to the length of time for which the backorder exists, and a fixed penalty cost is incurred per unit of lost demand. Based on a heuristic treatment of a lot-size reorder-point policy, a mathematical model representing the average annual cost of operating the inventory system is developed. The optimal operating policy variables minimizing the average annual cost can be calculated iteratively. At the extremes β = 1 and β = 0, the model presented reduces to the usual backorders and lost sales case, respectively.     

A Multi-Objective Production Inventory Model with Backorder for Fuzzy Random Demand Under Flexibility and Reliability

In this paper, an Economic Production Quantity (EPQ) model is developed with flexibility and reliability consideration of production process in an imprecise and uncertain mixed environment. The model has incorporated fuzzy random demand, an imprecise production preparation time and shortage. Here, the setup cost and the reliability of the production process along with the backorder replenishment time and production run period are the decision variables. Due to fuzzy-randomness of the demand, expected average demand is a fuzzy quantity and also imprecise preparation time is represented by fuzzy number. Therefore, both are first transformed to a corresponding interval number and then using the interval arithmetic, the single objective function for expected profit over the time cycle is changed to respective multi-objective functions. Due to highly nonlinearity of the expected profit functions it is optimized using a multi-objective genetic algorithm (MOGA). The associated profit maximization problem is illustrated by numerical examples and also its sensitivity analysis is carried out.     

Optimal ordering policies for periodic-review systems with replenishment cycles

In this paper, we consider inventory models for periodic-review systems with replenishment cycles, which consist of a number of periods. By replenishment cycles, we mean that an order is always placed at the beginning of a cycle. We use dynamic programming to formulate both the backorder and lost-sales models, and propose to charge the holding and shortage costs based on the ending inventory of periods (rather than only on the ending inventory of cycles). Since periods can be made any time units to suit the needs of an application, this approach in fact computes the holding cost based on the average inventory of a cycle and the shortage cost in proportion to the duration of shortage (for the backorder model), and remedies the shortcomings of the heuristic or approximate treatment of such systems (Hadley and Whitin, Analysis of Inventory Systems, Prentice-Hall, Englewood Cliffs, NJ, 1963). We show that a base-stock policy is optimal for the backorder model, while the optimal order quantity is a function of the on-hand inventory for the lost-sales model. Moreover, for the backorder model, we develop a simple expression for computing the optimal base-stock level; for the lost-sales model, we derive convergence conditions for obtaining the optimal operational parameters.     

Incorporating quantity discounts and their inventory impacts into the centralized purchasing decision

Multi-site organizations must balance conflicting forces to determine the appropriate degree of purchasing centralization for their respective supplies. The ability to garner quantity discounts represents one of the primary reasons that organizations centralize procurement. This paper provides methodologies to calculate optimal order quantities and compute total purchasing and inventory costs when products have quantity discount pricing. Procedures for both all-units and incremental quantity discount schedules are provided for four different strategic purchasing configurations (scenarios): complete decentralization, centralized pricing with decentralized purchasing, centralized purchasing with local distribution, and centralized purchasing and warehousing. For ordering decisions under local distribution, procedures to determine optimal order quantities and costs are presented in a precise form that could be easily implemented into spreadsheets by practicing managers. For the more complicated multi-echelon scenarios, we introduce a single-cycle policy with a tailored aggregation refinement step that performs very well under experimentation when compared to a conservative bound.     

A model for heuristic coordination of real life distribution inventory systems with lumpy demand

This paper presents an approximation model for optimizing reorder points in one-warehouse N-retailer inventory systems subject to highly variable lumpy demand. The motivation for this work stems from close cooperation with a supply chain management software company, Syncron International, and one of their customers, a global spare parts provider. The model heuristically coordinates the inventory system using a near optimal induced backorder cost at the central warehouse. This induced backorder cost captures the impact that a reorder point decision at the warehouse has on the retailers’ costs, and decomposes the multi-echelon problem into solving N + 1 single-echelon problems. The decomposition framework renders a flexible model that is computationally and conceptually simple enough to be implemented in practice.     

Economic Order Quantity and Quantity Discounts under Date-terms Supplier Credit: A Discounted Cash Flow Approach

This paper examines economic order quantity and optimal order quantity under both all-units and incremental-quantity discounts when purchase cost, order cost, and carrying cost are all incurred on date-terms supplier credit. Payment dates for the three cost components need not be the same. The traditionally-used periodic-cost minimization methodology which is insensitive to differences in the timing of various within-period cash flows is replaced with a discounted cash flow methodology. Differences in the characteristics of day-terms and date-terms solutions to the quantity discount case are high-lighted.     

A modified tabu search algorithm for cost-based job shop problem

In this paper, a cost-based job shop problem (JIT-JSP) is proposed to model the multi-order processing procedure in a just-in-time (JIT) environment. The objective of JIT-JSP is to minimize three costs: work-in-process holding cost of half-finished orders, inventory holding cost of finished orders and backorder cost of unfulfilled orders. A modified tabu search (MTS) method is developed to improve the schedule quality by searching the neighbourhood of a feasible schedule iteratively. The MTS method is comprised of three components that help to ensure a more effective searching procedure: neighbourhood structure, memory structure and filter structure. Computational results show that the MTS method significantly improves the initial schedule generated by an arbitrarily selected dispatching rule.     

Using aggregate fill rate for dynamic scheduling of multi-class systems

For dynamic scheduling of multi-class systems where backorder cost is incurred per unit backordered regardless of the time needed to satisfy backordered demand, the following models are considered: the cost model to minimize the sum of expected average inventory holding and backorder costs and the service model to minimize expected average inventory holding cost under an aggregate fill rate constraint. Use of aggregate fill rate constraint in the service model instead of an individual fill rate constraint for each class is justified by deriving equivalence relations between the considered cost and service models. Based on the numerical investigation that the optimal policy for the cost model is a base-stock policy with switching curves and fixed base-stock levels, an alternative service model is considered over the class of base-stock controlled dynamic scheduling policies to minimize the total inventory (base-stock) investment under an aggregate fill rate constraint. The policy that solves this alternative model is proposed as an approximation of the optimal policy of the original cost and the equivalent service models. Very accurate heuristics are devised to approximate the proposed policy for given base-stock levels. Comparison with base-stock controlled First Come First Served (FCFS) and Longest Queue (LQ) policies and an extension of LQ policy (Δ policy) shows that the proposed policy performs much better to solve the service models under consideration, especially when the traffic intensity is high.     

客户耐烦期相同的生产——库存系统最优模型

在常数需求率以及有限生产率条件下研究了订货客户耐烦期相同的一类新的生产——库存模型 ,在每一个周期内考虑了延期交货时间超过耐烦期的短缺费用和销售机会损失等因素 ,给出了相应的最优生产时间和周期的确定方法 ,利用数学软件 Matlab及计算机为工具给出了数字例子进行说明 ,其方法和结果为库存系统的管理决策提供了理论依据