Partial Backordering Research Articles

Evaluation of two transshipment policies in a two-location decentralized inventory system under partial backordering

This research investigates the replenishment and transshipment decisions in a two-retailer inventory system with a single selling season. Both emergency lateral transshipment (ELT) and preventive lateral transshipment (PLT) are investigated. ELT satisfies partial backordering. We prove the existence of unique Nash equilibrium for the system under each policy. The results show that the ELT solution is independent of the transshipment price while the PLT solution converges to the newsvendor solution as the transshipment price increases. Numerical analysis is presented to illustrate the advantage of each policy. This study assists retailers in deciding which transshipment policy should be preferred.

A unified EOQ model with financial constraints and market tolerance

• Deterministic lot sizing within the general EOQ with partial backorders framework is studied. • Financial constraints and tolerance to shortages are considered. • The unified profit is generally non convex and incorporates two separate branches. • Properties that lead to a closed form global optimal solution are provided. We consider the general EOQ (with partial backorders) paradigm to study factors related to the recent financial crisis. Upstream, there are financial constraints, which first limit the available capital per replenishment and second impose the prepayment (in full or partial) of the quantity ordered. Downstream, the concept of market tolerance is operationalized, permitting here backorders to occur at no-cost for a fixed time period (directly analogous to upstream payment credit). For the resulting system, we present an exact unified model of the total average profit/cost. Optimal profit/cost and respective decision variables (i.e. order cycle and shortage period) are subsequently obtained in closed form. Interestingly, by appropriately relaxing the solution of the unified system, the solution of several (some previously studied) EOQ-type models prevails. Finally, analytical and computational results reveal the impact that each factor (individually and jointly) has on system optimal solution and on the behavior of the respective optimizers.

Joint pricing and inventory decisions with carbon emission considerations, partial backordering and planned discounts

Typical economic order quantity models of inventory feature demand rate as a constant parameter and do not allow for backordering. Furthermore, the purchasing cost of the ordered materials is considered constant. In reality, the demand rate is related to the unit purchasing cost and other factors, such as time and availability of products in the market. A quantity discount is regularly applied to encourage ordering more products by decreasing the price. In some situations, carbon dioxide emissions are carefully scrutinized and a program to handle these. Greenhouse gases are put in place. Hence, for this research, the rate of demand in the model was assumed proportional to the unit purchasing cost and partial backordering was allowed as a fixed parameter. Because plants emit greenhouse gases (carbon dioxide), we considered mitigation efforts. A mathematical model and computational procedures are shown with the solution algorithms that demonstrate the capability of the model. An example problem was solved with the model and sensitivity analysis was conducted to inform the managerial insights offered.

Multi-item inventory model with variable backorder and price discount under trade credit policy in stochastic demand

In this paper, a two echelon supply chain with one manufacturer and one retailer is developed for multi products. The retailer faced with the uncertain demand for all products which follows a normal distribution. The production process is assumed to be imperfect, and the defectiveness is assumed to follow a beta distribution. The manufacturer produces and delivers the products in a number of equal-sized batches to the manufacturer's warehouse, and thereby it is delivers in a number of equal batches to the retailer's warehouse. Shortages are allowed to occur, at the retailer side, and it is backordered partially. The retailer offers a price discount for backordered items to his customers. Both the lead time crashing cost and the partial backorder ratio are considered as the inverse function of lead time. Under these assumptions, there are three inventory models proposed in this paper, one with non-integrated approach, the other with an integrated approach without trade credit and finally an integrated approach with trade credit. A new iterative algorithmic procedure has been developed to minimise the total cost. Finally, numerical examples are given to illustrate the models and the sensitivity analysis is conducted over various model parameters.

A sustainable supply chain for organic, conventional agro-food products: The role of demand substitution, climate change and public health

Abstract In today's marketplace, conventional agricultural products are produced all around the world due to economic reasons. However, recent environmental challenges such as global warming and climate change has strongly been affected by the high volumes of chemical pesticides and fertilizers used to produce these products. Several published reports have indicated the negative effects of the chemicals used in conventional agriculture as well as the effect of climate change on human health. The aim of this study is to develop and analyze a multi-objective linear mathematical model for a sustainable supply chain with an agro-food deteriorating product. The product is produced by two different methods, both by organic and conventional means. In this way, we attempt to create a balance between the production and consumption of conventional and organic products in order to achieve three objectives: reducing costs, reducing environmental degradation, and increasing levels of consumer health. Partial backorder for each product, as well as conventional demand substitution by organic products, are taken into account. In order to assess the impact of supply chain operations on public health, we define a new function based on each product's production and consumption amounts as well as the amount of wastes. The developed model is solved by augmented ɛ-constraint method. Numerical results indicate the important role of supply chain's managers in the formation of sustainable health production and consumption patterns.

Multi-objective optimization of multi-item EOQ model with partial backordering and defective batches and stochastic constraints using MOWCA and MOGWO

In this paper a multi-objective mathematical model is proposed for multi-item EOQ model considering partial backordering and defective supply batches. In order to consider real world situations, different stochastic operational constraints are considered and implemented under uncertainty. For the first time in this research, the rate of partial backordering is considered as decision variable. The aim is to determine time interval between successive supply deliveries, rate of partial backordering and filling rate from stock in order to minimize total inventory costs including holding, backordering and ordering costs, while, minimizing required warehouse space. Due to complexity and nonlinearity of the proposed mathematical model from one hand and importance of providing the decision maker with efficient Pareto optimal solutions, five meta heuristic algorithms as well as one hybrid exact solution method are utilized to solve the problem. To determine the most efficient solution method, the performance of the algorithms is investigated through a deep computational analysis considering different measures including diversity, number of Pareto optimal solutions, spacing and computation time. In addition, single factor ANOVA is utilized to determine significant difference among algorithms at ninety-five percent confidence level to demonstrate the superior algorithm.

Economic production quantity (EPQ) model with partial backordering and a discount for imperfect quality batches

Economic production quantity (EPQ) models are traditionally used in operations management. Despite the large number of papers that describe the models, the classic EPQ model does not consider either imperfect quality batches or shortages. However, some industries may be able to sell imperfect items for a lower price, reducing the total production cost. This paper proposes an EPQ model with partial backordering and discount for imperfect quality batches and an algorithm that returns optimal values for the problem. From a numerical example, it is possible to analyse how the changes in the variables affect each part of the total cost function, which provides a useful tool for strategic decision-making. We conclude that it is better to sell imperfect items as soon as possible because the savings in holding costs results in a total cost reduction. It is more profitable for the producer to have planned shortages considering that some costumers are willing to wait. Furthermore, the reduction of the goodwill cost does not necessarily reduce the total cost.

An integrated inventory model with variable holding cost under two levels of trade-credit policy

This paper presents an integrated vendor-buyer model for deteriorating items. We assume that the deterioration follows a constant rate with respect to time. The vendor allows a certain credit period to buyer in order to promote the market competition. Keeping in mind the competition of modern age, the stock-dependent demand rate is included in the formulated model which is a new policy to attract more customers. Shortages are allowed for the model to give the model more realistic sense. Partial backordering is offered for the interested customers, and there is a lost-sale cost during the shortage interval. The traditional parameter of holding cost is considered here as time-dependent. Henceforth, an easy solution procedure to find the optimal order quantity is presented so that the total relevant cost per unit time will be minimized. The mathematical formation is explored by numerical examples to validate the proposed model. A sensitivity analysis of the optimal solution for important parameters is also carried out to modify the result of the model.

Inventory parameters for a serial supply chain with lost sales through genetic algorithm approach

Supply chain is a network of facilities with varying conflicting objectives and decision making is a complex process. The uncertainty in demand increases complexity in inventory control mechanism. Backordering, partial backordering and lost-sales are considered in the inventory management to characterise the excess demand. In the present competitive scenario, mostly consumers have no patience to wait and show urgency to buy their goods failing which the management has to meet a huge loss of supply chain members and hence the profit. There are very few research works regarding lost-sales parameter in the area of multi-echelon inventory systems. It is felt that the proposed modified gene-wise genetic algorithm (MGGA) supply chain model which so far not applied in this process may help to determine best base stock levels and review periods with lost sales particularly at retailer end which minimised total supply chain cost.

Two ware-house Fuzzy Inventory model for Deteriorating Items with Ramp Type Demand and Shortages

In this paper a fuzzy inventory model for a single deteriorating itemwith two parameter weibulldistribution deterioration rate, ramp type demand and partial backordering at constant rate is developed. In the present market scenario due to inflation an increasing cost ofvarious factors affecting the total worth of inventory cost may increase ordering cycle length. Increase in the cost of components of inventory may not be pre-determined due to uncertainty of the market situation. Therefore, an interval based fuzzy concept has been considered to handle the situation of uncertainty. Orderingcost, holding costs in both ware-houses are assumed to be triangular fuzzy numbers. The objective of this paper is to derive an optimal replenishment policy with defuzzification of fuzzy numbers using signed distance method to minimize the average inventory cost. Further numerical examples are presented to illustrate the model. Initiallya crispmodel has been developed and then corressponding fuzzy model is derived separately and the results of twomodels are compared and analysed.

Inventory parameters for a serial supply chain with lost sales through genetic algorithm approach

Supply chain is a network of facilities with varying conflicting objectives and decision making is a complex process. The uncertainty in demand increases complexity in inventory control mechanism. Backordering, partial backordering and lost-sales are considered in the inventory management to characterise the excess demand. In the present competitive scenario, mostly consumers have no patience to wait and show urgency to buy their goods failing which the management has to meet a huge loss of supply chain members and hence the profit. There are very few research works regarding lost-sales parameter in the area of multi-echelon inventory systems. It is felt that the proposed modified gene-wise genetic algorithm (MGGA) supply chain model which so far not applied in this process may help to determine best base stock levels and review periods with lost sales particularly at retailer end which minimised total supply chain cost.

Entropic Order Quantity (EnOQ) Model for Decaying Items with Partial Backordering and Lost Sale

Entropic Order Quantity (EnOQ) Model for Decaying Items with Partial Backordering and Lost Sale

Ordering policies for non-instantaneous deteriorating items under hybrid partial prepayment, partial trade credit and partial backordering

In this paper, an economic order quantity model is presented for non-instantaneous deteriorating items under a hybrid payment schedule. This payment schedule is composed of a multiple advanced payments scheme and a delayed payment plan. Here, a retailer must prepay a portion of the purchasing cost to his supplier, during the lead time of order delivery, in several instalments. The retailer is allowed to pay the rest after a certain amount of time from receiving the order. The first policy may be adopted by the supplier to finance the procurement of materials or parts used to prepare the order, or to control the risk of order cancellation; and the second policy may be employed as a marketing strategy to stimulate sales. On the other hand, the retailer sells the products to his customers. For the proposed model, inventory shortage is also taken into account, which may occur as either backordered and lost sales, or both. The retailer’s total inventory costs (including the costs of ordering, purchasing, inventory holding, shortage and also the interest costs incurred for advance payment and delayed payment) is minimised, in order to find the order and shortage quantities. Several numerical examples are presented for demonstrating the applicability of the framework. Finally, in order to provide managerial insights, sensitivity analyses are performed for several key parameters.

Optimizing setup cost in (R, T) inventory system model with imperfect production process, quality improvement, and partial backordering

Managers always want to decrease the cost of the inventory system and increase the customers’ satisfaction. To access to the mentioned aim they should notice the different aspects of their inventory system to become successful in today competitive global marketing. In this research we model the periodic review inventory system by considering some important aspects of the system that are improving quality, reduction of setup cost, partial backordering and inspection process. The provided model is the first one that includes all of the mentioned aspects simultaneously. Finally, we provide the numerical result according to the certain algorithm and show how the sensitive parameters affect the total cost. We compare the total cost in two cases: at first, we consider setup cost and the probability of the production that results in producing defective items as parameters and in the second case we consider them as variables.

Sustainable economic production quantity models for inventory systems with shortage

Recently, new economic order/production quantity models have shifted away from focusing only on economic issues and towards combined economic-environmental concerns because of sustainable development goals. Despite this shift, only a few works have addressed sustainable Economic Production Quantity (EPQ). The theoretical sustainable EOQ and EPQ models are basic models that ignore many real-life conditions such as the possibility of stock-out in inventory systems. In this paper, we develop four new sustainable economic production quantity models that consider different shortage situations. To find optimal values of inventory system variables, we solve four independent profit maximization problems for four different situations. These proposed models include a basic model in which shortages are not allowed, and when shortages are allowed, the lost sale, full backordering and partial backordering models can be selected by operations managers depending on the manufacturer's motivation to improve service levels. We have also proposed an algorithm for determining optimum values of the decision variables for these sustainable economic production quantity models. Finally, the formulated models are explained with some different examples and the obtained results have been analyzed and discussed. These results show that the sustainable economic production quantity with partial backordering model is a general and more realistic model that can be used in many real cases with a reasonable profit amount, compared with the three other proposed models.

A lot sizing model with partial downstream delayed payment, partial upstream advance payment, and partial backordering for deteriorating items

In practice, installment of cost of lots is very common between manufacturers and distributors, and many distributors pay the purchasing cost in equal installments (i.e., an upstream partial prepayment). In a similar fashion, distributors allow retailers to pay the cost of purchased goods after such goods are received (i.e., a downstream partial delay payment). In this research we apply a model of economic order quantity (EOQ) in supply chains with partial downstream delayed payment and partial upstream advance payment for a deteriorating item under three conditions: 1) shortage is not allowed, 2) full back ordering is allowed, and 3) partial back ordering is allowed. We prove the convexity of the cost functions and derive closed form global optimal solutions for the decision variables of all models. Eventually, to demonstrate the theoretical results and managerial options, we perform some computational examples.

An EOQ model with partial backordering with regard to random yield: two strategies to improve mean and variance of the yield

This paper presents an economic order quantity (EOQ) inventory model with partial backordering, where a buyer purchases its required products from a supplier. Each received batch from the supplier includes a random proportion of defective items. In other words, supplier’s production process works to a random yield. A recursive solution method to find the optimal values for main decision variables is examined through a series of numerical examples. Moreover, it is considered that the yield can be improved at a given cost. To this extent, two ways to achieve the yield improvement have been considered: the first way is investment in order to improve the mean of the yield to an ideal value, and the second way is investment to reduce the yield variability to an acceptable value. The impacts of the proposed strategies are investigated through some numerical analysis and managerial insights are proposed.

A Partial Backordering Inventory Model with a Drop-Shipping Option Under Purchase Dependence

Drop-shipping is a commonly adopted online-order fulfillment strategy in the Internet age. In this practice, online retailers leverage the fulfillment capabilities of suppliers to fulfill orders. On the other hand, purchase dependence is a frequent phenomenon and is characterized by the purchase of certain items together due to their unknown interior associations. Although this concept has been significantly examined in the marketing field (e.g., market basket analysis), it has largely remained unaddressed in operations management. This paper develops an [Formula: see text] model to address an environment in which unmet demand orders are partially lost and partially backordered when purchase dependence exists. The partial backorders are fulfilled by a drop-shipping option. Through computational analyses, this paper demonstrates the effect of both drop-shipping on a partial backordering and purchase dependence. The results show that more profit can be realized by utilizing a drop-shipping option under purchase dependence.

Stochastic Multi-Objectives Supply Chain Optimization with Forecasting Partial Backordering Rate: A Novel Hybrid Method of Meta Goal Programming and Evolutionary Algorithms

This study proposes a model for a multi-objective, multi-buyer, multi-vendor, multi-product and multi-constraint supply chain. The buyers’ demand rates are stochastic variables with known probability distribution functions. The classical ([Formula: see text]) inventory control system is used to manage the inventories of all buyers where the lead times are production rate dependent. Shortage is permitted and is partially backordered where the partial back ordering rate is forecasted. The model is considered as a multi-objective integer nonlinear programming problem including cost, service level and lead time objectives and using a novel hybrid method, a hybrid of Meta Goal Programming (MGP) and Firefly Algorithm (FA) are solved. Numerical examples are given to illustrate the proposed method in the study. The results of the study are compared to other hybrid methods of Meta Goal Programming with other evolutionary algorithms such as Bees Algorithms (BA), Particle Swarm Optimization (PSO), Genetic Algorithm (GA) and Simulated Annealing (SA).

Optimizing stock levels for rental systems with a support warehouse and partial backordering

Various rental systems, such as public libraries and tool rental companies, have a support warehouse for storing rental products at low cost and carrying out expedited shipments in response to stock-outs at local warehouses. In case of stock-outs, rental systems commonly employ partial backordering, i.e., a limited number of demands is backordered while additional demand is lost. We include partial backordering in our models and demonstrate its relevance. We fully characterize costs and optimal base stock policies in cases with one support and one local warehouse and provide upper bounds for base stock levels in the general system. By enumerating base stock policies and evaluating exact costs numerically, we determine optimal base stock policies for small systems with up to six local warehouses. For larger systems we develop a greedy algorithm based on approximate cost evaluations. The resulting base stock policies have costs within 0.2% from optimality on average in experiments. Among other things, experiments indicate that existing methods with no backordering may lead to poor solutions with partial backordering and that partial backordering is an effective means to reduce lost demand even with limited partial backorder levels.