Partial Backordering Research Articles

Optimal order quantity considering carbon emission costs, defective items, and partial backorder

To respond to the adoption of carbon pricing regulations, researchers and industry are developing low carbon inventory models that can meet emission reduction targets while maintaining company profits. The challenge is getting tougher when the company is still facing problems related to imperfect product quality. This research solves this problem by developing an economic order quantity (EOQ) model by considering several sources of carbon emissions, as well as the influence of the defective rates, different demand rates, selling price and holding cost for defective products, and shortages backorder. The objective function of the formulated mathematical model is to minimize the total costs which include the emission costs. A numerical example is developed to illustrate the model based on the previous data set. Sensitivity analysis is also carried out to validate the model and to learn more about the system characteristics. The total emissions are calculated and the affecting factors are identified.

Preservation technology investment, trade credit and partial backordering model for a non-instantaneous deteriorating inventory

In a perfectly transparent and competitive market, suppliers must provide a competitive pricing and service for their customers. The aim of this study is to provide an insight into how preservation technology and credit financing could be used both to reduce the deterioration rate as well as to provide flexible financing for retailers. The methodology is to optimize the cycle length, selling price, the amount of preservation technology and credit financing using inventory theory. The result derived is an optimal total profit per unit time for the system. Finally, using MATLAB 2017a, it is shown graphically that the profit function is concave. The sensitivity analysis is illustrated using Lingo 17. The study not only provides insights to business managers in making wise managerial decisions, it also enables them to weigh the pro and con of implementing preservation technology and credit financing.

Inventory policies for seasonal items with logistic-growth demand rate under fully permissible delay in payment: a neutrosophic optimization approach

The present study investigates an inventory system for seasonal products under variable demand rate and partial backordering in a competitive market. Among various demand rate functions used in the existing literature of economic order quantity (EOQ) models, the logistic-growth function is best known to estimate market already captured and fraction of market remaining to be captured by new seasonal and technology driven items. Weibull distribution well represents the seasonality and versatility of these products. Due to retailers’ reluctance to purchase and store these perishable products, supplier offers the delay in payment. In view of the above, the proposed EOQ model suitable for those items considers the logistic-growth demand rate, Weibull distribution deterioration rate and partial backordering along with fully permissible delay in payment. Since the neutrosophic set quantifies the imprecise information in real-life scenarios, the proposed EOQ model is optimized in neutrosophic environment. A general unconstrained nonlinear mathematical model with neutrosophic coefficients is optimized using the weighted arithmetic mean function, subject to specified neutrosophic norm. A special case with the neutrosophic conjunction norm along with four lemmata is considered to minimize the cost functions with neutrosophic coefficients to proposed EOQ model across various trade-credit intervals. Here, the managerial insights identified through sensitivity analysis advocate to reduce the expenses on early promotions for foreshortening the demand at the beginning of cycle. Also, the present study demonstrates the optimal inventory depletion time to depend on the demand during the shortages in neutrosophic environment.

EPQ models for an imperfect manufacturing system considering warm-up production run, shortages during hybrid maintenance period and partial backordering

Unreliable conditions and process deterioration of machines are the major issues in several manufacturing systems. These issues not only affect the production schedules but also affect the product quality significantly. A process warm-up production and amalgamation of the preventive and corrective maintenance are important for the smooth functioning of production activities and improving the product quality. The management needs to handle the product shortages carefully and raise the backorders to upgrade the business economy. To accomplish all these objectives, we develop two production inventory models in this paper by considering a process warm-up production run, hybrid maintenance schedule, anticipated shortages during the maintenance period, and rework of imperfect items. Two different rates are used to distinguish the production of imperfect items in the warm-up and standard production periods. We use fixed and variable lengths to represent the warm-up production period in the two models respectively. The maximum customer waiting time for receiving the backorders is calculated under the consideration of finite replenishment policy and simultaneous regular demand satisfaction. Numerical examples and sensitivity analysis are provided to illustrate the models. The two models are compared based on their annual variable costs. The results suggest that a trade-off between the rework and holding costs of defective items is necessary for better inventory decisions. Determining the optimal value of the length of the warm-up production period is necessary to minimize the annual variable cost.

Synergic Effect of Reworking for Imperfect Quality Items with the Integration of Multi-Period Delay-in-Payment and Partial Backordering in Global Supply Chains

Abstract In intercontinental trade and economics goods are bought from a global supplier. On occasion, the expected lot may include a fraction of defective items. These imperfect items still have worth and can be sold to customers after repair. It is cost-effective and sustainable to rework such items in nearby repair workshops rather than return them. The reworked items can be returned from the workshop to the buyer when shortages are equal to the quantity of imperfect items. In the meantime, the supplier correspondingly deals a multi-period delay-in-payments strategy with purchaser. The entire profit has been maximized with paybacks for interim financing. This study aims to develop a synergic inventory model to get the most profit by making an allowance for reworking, multi-period delay-in-payments policy, and shortages. The findings of the proposed model augment inventory management performance by monitoring cycle time as well as fraction of phase with optimistic inventory for a supply chain. The results demonstrate that profit is smaller if the permitted period given by supplier to buyer is equal to or greater than the cycle time, and profit is greater if the permitted period is smaller than the cycle time. The algebraic method is engaged to make a closed system optimum solution. The mathematical experiment of this study is constructed to provide management insights and tangible practices.

Resilience toward supply disruptions: A stochastic inventory control model with partial backordering under the base stock policy

This paper examines supply-side disruptions using an inventory management framework to gain insights into the economic performance of buyers who use two conventional costing strategies – end-of-cycle and continuous. The proposed model assumes that a single supplier faces full disruptions with a probability, and therefore, fails to procure all the items ordered. Accordingly, the buyer experiences unmet demand, which is assumed to be partially backordered. To make proper replenishment decisions, two base stock (S,T) periodic review optimization models are developed. The objective functions minimize the expected long-run total costs (i.e., ordering, holding, and shortage). After proving their convexity and obtaining the optimal decisions, computational experiments were carried out to investigate the impacts of the change in parameters. The results highlight the importance of selecting a suitable ratio for backorders during supply disruptions, as it leads to lower costs and reduces the inventory obsolescence and overstocking risks. Moreover, the managerial insights derived from this study aid retailers to make better replenishment decisions (in terms of level and frequency) and to be more resilient in times of disruptions. Retailers can benefit from the provided solution algorithm as a computational application.

Optimum sustainable inventory management with backorder and deterioration under controllable carbon emissions

Reducing carbon emissions is crucial in the development of a green supply chain. Greenhouse firms regulate carbon emission constraints to achieve sustainable supply chains. Deterioration can be controlled by utilizing a preservation technology investment in most of greenhouse firms as they have a specific (summer) time. Despite the importance of these concerns, few researches have been done on the effect of carbon emissions on the lot sizing sustainable inventory management. To fill this gap, this study develops a carbon cap and tax-regulated sustainable inventory management for a buyer utilizing a linear and non-linear price-dependent demand. This is the first study to explore sustainable inventory management under controllable carbon emissions from a greenhouse farm. An investment in preservation and green technology efforts under different backorder situations is needed. This model furnishes both theoretical and numerical solutions to determine optimal strategies relating to preservation technology (PRT) and green technology (GRT) investment, selling price, cycle time, and the fraction period length for sustainable inventory management. Several numerical examples are presented to validate all sustainable economic order quantity models. This problem is solved by theoretical and analytical methods. The results of this paper, in a controllable sustainable inventory management partial backordering case, exhibit a justifiable amount of profit compared with other backordering cases. The results of this paper are sustainable inventory management in the carbon tax and cap partial backlogging case has a better justifiable profit with the highest cycle time and the lowest value of the fraction length period as well as the lower green technology investment cost compared with other models. Finally, sensitivity analysis is discussed in relation to the managerial implications of the optimal feasible solution with respect to key parameters.

An inventory model under price and stock dependent demand for controllable deterioration rate with shortages and preservation technology investment: revisited

Mishra et al. (Ann Oper Res 254(1–2):165–190, 2017) proposed an EOQ model for a deteriorating seasonal product where demand was considered as a function of stock and selling price. Their model practiced preservation technology in order to control the deterioration rate. Further, shortages were allowed and two different scenarios, complete backordering and partial backordering were dealt with in individual cases. The model jointly optimized selling price, ordering frequency and investment in preservation technology under a profit maximization scenario. The problem they investigated depicts a practical real-life scenario, but the mathematical modelling is incorrect. Thus, the numerical results, special cases and managerial insights are debatable. Realizing the significant contribution of the model in question to the existing literature, we revisit the model to provide a corrected comprehensive version. The present model’s optimality has been proved mathematically. Further, the model has been validated using numerical analysis. Sensitivity analysis has been performed to test the robustness of the model.

An Extension to the Economic Production Quantity Problem with Deteriorating Products Considering Random Machine Breakdown and Stochastic Repair Time

The recent advances in manufacturing systems motivate several studies to focus on Economic Production Quantity (EPQ) problem. Althuogh there are several extentions to the EPQ, this paper provides a new extension by considering some of the real world parameters like: (a) shortages in the form of partial backordering, (b) inventory can deteriorate stochastically, (c) machine can break down stochastically, and (d) machine repair time may change stochastically based on the failure status of machine. As far as we know, there is no study treated all these suppositions in an EPQ framework. In addition to this development, two forms of uniformly- and exponentially-distributed repair times are formulated and necessary convexity conditions are discussed. Then, the corresponding optimality conditions are written that lead to finding the roots of two equations. Due to difficulty of achieving a closed-form solution, the solution is obtained numerically by means of Newton-Raphson method. Finally, some sensitivity analyses are provided to explain the models’ applicability. The practicality and efficiency of the proposed method in this context lends weight to development of proposed EPQ with more complex elements and its application more broadly.

Stackelberg scheme in vendor – buyer inventory model for imperfect quality product with partial backordering

In this paper, several new mathematical analysis approaches in probabilistic inventory model which incorporate both imperfect quality items existence and lead-free demands are proposed using non-cooperative games. Considered the monopolistic relationship between vendor and buyer in some real inventory problem, the vendor is set to be a leader who determines the Stackelberg’s optimal move after observing the optimal responses from the buyer as a follower. Optimum condition using Karush – Khun – Tucker conditions is applied with Stackelberg scheme to get optimum value of the system.

Stock replenishment policies for a vendor-managed inventory in a retailing system

Using Vendor Managed Inventory (VMI), the vendor determines the replenishment decisions at the location of buyers (retailers). This strategy is used primarily for handling demand fluctuations stemming from the Bullwhip effect, leading the system to prevent from holding excessive inventory that result in a reduction in the overall cost of the supply chain. The main advantages of VMI for vendors are higher levels of accessibility to inventory information and more direct contact with the customers. Similarly, VMI has some pros for the buyers, such as shared risk with upper levels of supply chain and reduction in their holding costs of inventory. In this paper, a vendor-managed inventory system is developed containing one vendor and two buyers in which the main assumption is that back-ordering and lost sales are permitted. In this system, (r, Q) and (R, T) replenishment policies are compared according to their performances to see which one performs more cost-efficiently when partial back-ordering is allowed. In accordance, mathematical models utilizing (r, Q) and (R, T) replenishment policies are developed, and algorithms for deriving the optimal replenishment decision variables are proposed. Moreover, significant differences between the two replenishment policies are discussed. The main finding obtained by this research is that when shortage is permitted, both (r, Q) and (R, T) replenishment policies under VMI have pros and cons in different contexts.

Effects of Unequal Lot Size and Variable Transportation in Unreliable Supply Chain Management

The effect of unreliable players on the supply chain management with a single-setup-multi-unequal-increasing-delivery-policy (SSMUID) along with a service-dependent demand and investment is discussed in this model. The manufacturer is unreliable which causes an increase of lead time and shortage. For solving the shortage problem and reducing lead time crashing cost (LTCC), an investment is utilized with the variable backorder price discounts. The number of transportation increases due to the new transportation policy and it causes pollution. Besides the fixed transportation and carbon emission cost (FTCEC), a container dependent carbon emission cost is applied. Some investments for setup cost reduction (SCR), ordering cost reduction (OCR), and quality improvement (QI) are considered. The lead time demand follows a normal distribution. The total cost of the supply chain is optimized and the model is tested numerically. The main intent of this study is to solve the shortage problem which occurs due to unreliability of the manufacturer. The study helps to reduce the unreliability issue of the manufacturer. The objective function is solved by using the classical optimization technique. Numerical results show that the discount for partial backorder enhances the profitability of the manufacturer. The sensitiveness of the parameters are discussed through the sensitivity of analysis and some special cases. Managerial insights provide the applicability of this study among different sectors.

Replenishment of imperfect items in an EOQ inventory model with partial backordering

This paper deals with an inventory model in which a percent of the items in the lot is imperfect. The supplier is far from the buyer. After the reception of the order, immediately the products are inspected and imperfect items are identified. Due to the fact that supplier is located at long distance and the demand is needed to cover, the imperfect items are replenished by perfect ones from a local supplier at higher cost. In addition, the imperfect items are withdrawn and sold at a salvaged price as second-degree items. The shortage is allowed and partially backordered. The following three cases are considered: Case I. The reordered items are received when inventory level is zero; Case II. The reordered items are received when the backordered quantity is equal to the imperfect items quantity; and Case III. The reordered items are received when shortage is still remained. These cases are studied and analyzed in detail. In each case, the aim is to obtain the optimal value of the length period and the percent of period duration in which the inventory level is positive. A numerical example is presented to show the applicability of proposed inventory model. The results show that Case I has the lowest holding and shortage cost, so the total benefit is higher than the other two cases.

A sustainable production-inventory model for a controllable carbon emissions rate under shortages

Carbon emissions are the key factor of global warming. Nowadays, some countries have focused on reducing carbon emission. Carbon tax and cap policies are the key mechanism utilized in many countries to achieve this. Keeping this issue in mind, a sustainable economic production quantity (SEPQ) carbon tax and cap model has considered for a controllable carbon emissions rate by investing in green technology (GT) efforts under different with and without shortage situations. This study has three models; a) a sustainable economic production quantity carbon tax and cap model with no shortages; b) a sustainable economic production-quantity carbon tax and cap with partial backordering; and c) a sustainable economic production-quantity carbon tax and cap with full backordering with and without green technology investment. A solution methodology has been proposed for determining the optimal strategies of cycle time, green technology investment, and the fraction period length in positive inventory level. Several numerical examples have studied to validate the sustainable economic production quantity models. The results of this paper are a sustainable economic production quantity carbon tax and cap partial backlogging case has a better justifiable profit with the highest cycle time and the lowest value of the fraction length period as well as the lower green technology investment cost compared with other models. Finally, sensitivity analysis has provided with the managerial implication of the optimal feasible solution with respect to some key parameters.

Optimal ordering policy for complementary components with partial backordering and emergency replenishment under spectral risk measure

A firm (assembler) faces random demand for a final product which is made up of multiple complementary components. Before demand is realized, the firm purchases the components via a regular channel. After demand realization, unsatisfied demand is allowed to be partially backordered in case of component shortages, where the firm is assumed to have an option to purchase the components via an emergency channel with a relatively higher unit cost. The firm needs to adopt an appropriate ordering policy to maximize the spectral risk measure of its profit. We formulate and transform the newsvendor problem into a concise optimization problem which can be further decomposed into two sub-problems. We provide optimality properties and show that the objective function of each sub-problem is separable, which enables us to determine the optimal order quantity of each component independently. We show that the optimal order quantity of each component decreases in the firm's level of risk-aversion. Numerical experiments are conducted to illustrate the effectiveness of our solution method and examine the impacts of lost sales and risk attitude on the optimal solutions.

Optimizing vendor-managed inventory systems with limited storage capacity and partial backordering under stochastic demand

A supply chain member’s coordination is a challenging issue and a key factor of success in business markets. In vendor-managed inventory systems, the vendor makes replenishment decisions at the site of buyer by which the supply chain can be coordinated more efficiently. Two integrated vendor managed inventory systems under continuous review and periodic review replenishment policies are developed considering partial backordering and limited storage capacity at the buyer’s side. Furthermore, traditional retailer managed inventory systems under the same settings are developed to compare against the integrated systems. Efficient algorithms are presented to derive the optimal values of decision variables. Finally, numerical experiments and comprehensive sensitivity analysis are used to show the applicability and efficiency of the proposed VMI systems.

Derivation of closed-form expression for optimal base stock level considering partial backorder, deterministic demand, and stochastic supply disruption

In the presence of stochastic supply disruption, the optimal variables of an inventory policy must be determined appropriately. Considering a two-echelon system comprised of a supplier and a retailer, the objective of this research is to help the retailer derives the optimal base stock level that achieves the minimum costs per unit of time regarding the stochastic unavailability of the supplier. The expression of the optimal base stock level is determined in closed-form in consideration of a continuous random variable of a disruption length together with a partial backorder of shortage inventory. A solution method which facilitates the retailer to derive the correct expression for the optimal base stock level is proposed. The applicability of the proposed solution method is illustrated through numerical experiments.

Inventory and pricing decisions for imperfect quality items with inspection errors, sales returns, and partial backorders under inflation

In this paper, an optimal replenishment inventory policy for imperfect quality items is presented with a selling price-dependent demand under inflationary conditions using a discounted cash flow (DCF) approach. Due to the presence of defectives in the system, all items go through a 100% inspection process. However, the screening process is also considered to be imperfect and involves errors, namely Type-I and Type-II. In addition, shortages are allowed and are partially backlogged. An optimal solution for the proposed model is derived by maximizing the expected profit function by jointly optimizing three decision variables: selling price, order quantity, and backorder level. To validate the theoretical results, a numerical example along with comprehensive sensitivity analysis is offered. The model has pertinence in industries like textiles, electronics, furniture, footwear, automobiles, and plastics.

AN OPTIMIZATION OF FEOQ MODEL FOR WEIBULL DETERIORATING ITEMS WITH INFLATIONARY CONDITION

The economic instability is a haunting situation in the present era for the economy. There is a monetary depreciation due to the tremendous increase in the price of the commodities. Present study basically deals with the investigation of the inventory system considering the case of perishable products under inflationary environment under condition of partial backorder. The rate of demand is a function which exponentially increases with increase in inflation and deterioration is taken as a Weibull distributed two parameter function. In the state of shortage when the inventory system runs out of stock an assumption is made that the demands are backlogged or lost. There is a variation in the rate of backlogging which shows this variation as a consequence of lead time required for the arrival of the next replenishment. The deterioration of commodities begins after a fixed interval of time. Present model aims at minimizing the average total cost considering both the crisp and fuzzy environments considering both inflationary effect as well as the time value of money. For the fuzzy economic order quantity (FEOQ) model of the inventory system fuzzification has been done by the use of fuzzy number of the trapezoidal nature. Defuzzification has been done by the use of centroid method. The primary motive of this work is to determine and compare the cost in total of the system of inventory in both crisp and fuzzy environments. The main inference that we can draw from this study is that fuzzy economic order quantity (FEOQ) model has more accuracy as in this model the total cost has been reduced as compared to the crisp model. Thus fuzzy economic order quantity (FEOQ) model is highly beneficial to any sort of inventory system and this result can be further generalized to any sector for enhancing the total profit of the system. Appropriate example has been provided for the illustration of both the models.

A production inventory supply chain model with partial backordering and disruption under triangular linguistic dense fuzzy lock set approach

This paper focuses on a three-level distribution process in a supply chain (SC) modeling where the raw materials are received batchwise with imperfect quality. Defective batches are rejected instantly under “all or none” policy. Allowing partial backlogging and random disruption in supply, we develop an expected average cost function of the production inventory SC model first. Then, considering the several cost components of the model as linguistic triangular dense fuzzy lock set, the cost function itself has been fuzzified according to the needs of the problem defined at case study. Utilizing the proper application (growth) of key vectors, the objective function has been solved under crisp, general fuzzy, dense fuzzy, dense fuzzy lock of single and double keys environment, respectively. For managerial importance, numerical results and graphical illustrations are made to justify the novelty.