Allowable Shortage Research Articles

Expected value of generalized trapezoidal bipolar fuzzy number to solve a multi-item marketing planning inventory model with allowable shortages

Decision makers often grapple with the complexity of multi-faceted real-life problems, where uncertainty arises from both positive and negative states of mind. This duality reflects the decision maker’s optimistic and pessimistic perspectives during the decision-making process. To address this, our paper introduces innovative concepts and frameworks for generalized bipolar trapezoidal fuzzy numbers.We propose novel arithmetic operations on these fuzzy numbers, computing the negative α-cut and positive β-cut methods. Furthermore, we uniquely compute the convex combination of expected values from both the positive and negative membership parts.These theoretical advancements are applied to a practical case study: a multi-item marketing planning inventory model with allowable shortages. Our proposed method’s efficacy is highlighted through detailed numerical illustrations, sensitivity analyses and comparative studies complemented by compelling graphical presentations

A Bi-Objective Inventory Model for Strategic Items in a Stochastic Environment with Partial Shortage: A Base-Stock Approach

This paper suggests an ordering policy for strategic items. Based on the base stock policy, we modeled a bi-objective, multi-product inventory control system with partial back-ordering. The model aims to minimize total costs and maximize the average service level, considering stochastic constraints such as storage space, maximum allowable shortage, and maximum allowable lost sales to match real-world conditions. We used three Multi-Objective Decision-Making methods, namely Individual optimization, Desirability function, and Goal attainment, to solve the problem. TOPSIS method showed the Goal attainment method’s superiority over the others. The sensitivity analysis revealed that the mean of storage space had the most significant impact on objective functions. Also, increasing the back-order percentage can reduce the system’s total costs.

Optimal pricing and advertisement with allowable shortages for non-instantaneous deteriorating items under inflation and trade credit

Optimal pricing and advertisement with allowable shortages for non-instantaneous deteriorating items under inflation and trade credit

A Multi-Objective Model and Algorithms of Aggregate Production Planning of Multi-Product with Early and Late Delivery

Due to the influence of insufficient production capacity or shortage of production materials, production enterprises may produce products in advance or be backordered. In order to improve the adaptability of enterprises and reduce production costs, the impacts of early delivery and delayed delivery are analyzed, and the method to determine the loss threshold is put forward. Moreover, the maximum allowable shortage of customers with different tardiness is calculated, and the cost of delayed delivery and loss of sales is determined. Considering the production cost, raw material cost, inventory cost, staff cost, stockout, and lost sales cost, an early/delay multi-objective optimization model is developed for an aggregate production planning (APP) problem to minimize total production costs and instability in the workforce. Three algorithms and three different hybrid strategies are designed to solve the model. Finally, some test experiments are employed in order to validate the performance of the proposed evaluation of the three algorithms. The results show that: The method of determining the loss threshold can effectively reflect the double influence of customer satisfaction with waiting time and shortage quantity. The definition of unit tardiness cost reflects the law that it increases gradually with waiting time. The determination of the feasible range of product output and the number of workers in the workforce can reduce the search scope of the algorithm and improve the efficiency of the algorithm.

Inventory management with hybrid cash-advance payment for time-dependent demand, time-varying holding cost and non-instantaneous deterioration under backordering and non-terminating situations

During the current pandemic period, the number of online transactions to purchase products through a mixed cash and prepayment agreement has increased significantly. Under this agreement, the buyer pays off a predefined percentage of the purchase price in advance, and the rest of the percentage is accomplished through the cash-on-delivery method. Adopting a practical scenario where the buyer has a storeroom with limited capacity, two sustainable inventory procedures for perishable items are developed, viz. (i) the inventory scheme with allowable shortages and fractional backlog, and (ii) the inventory scheme without ending. In both the schemes, not only the demand function but also the decay rate reveal an upward trend against storage time. However, the deterioration commencing moment is not instant but after a specific time interval from the products’ storage moment. The cost of storing items is adopted as a linear function of storing time as the decay rate shows an ascending trend against storing time. A salient criterion is developed to examine when the non-ending situation with salvage values is more economical for the industry manager than the zero ending situation with the regular selling price. Finally, several decision-making findings are presented after completing four numerical examples along with a real-life case study for the validation of the inventory schemes.

Joint inventory, promotion and preservation decisions for deteriorating items with maximum lifetime and stochastic demand under two-level partial trade credit

This study models a joint inventory, promotional effort and preservation decision-making problem for deteriorating items with maximum lifetime under a two-level partial trade credit and allowable shortages. This paper considers a supplier-retailer-customer supply chain model which allows: 1) for settling the cost of purchasing, the supplier offers a partial trade credit to the retailer and at the same time retailer offers a partial trade credit to the customer; 2) the upstream(supplier-retailer) credit period increases sales of the supplier and revenue of the retailer, the downstream (retailer-customer) credit period not only lifts demand but also the opportunity cost; 3) the deteriorating product not only deteriorate continuously, but also have maximum lifetime and to reduce the deterioration rate we use the preservation technology; 4) price and promotional effort are dependent on random demand; 5) shortages are considered. The objective is to find the optimal promotional effort, preservation technology investment, length of time for the inventory level reaches zero and replenishment cycle strategies while maximising the total profit per unit time. Numerical examples are included to illustrate the algorithmic procedure and the effect of key parameters is studied to analyse the behaviour of the model.

Fuzzy optimization model of two parameter Weibull deteriorating rate with quadratic demand and variable holding cost under allowable shortages

In this paper, a fuzzy inventory model with a Weibull deterioration rate, a quadratic demand rate, and a variable holding cost under permissible shortages has been developed. The deterioration rate is expressed by a two-parameter Weibull distribution. During a shortage, some buyers wait for the actual product, while others do not. This shortfall is considered partially backlogged in this model. Some buyers wait for the actual product during such shortages, but many do not. Therefore, partially backlogged shortages are taken into account in this approach. In a traditional inventory model, all parameters such as purchasing cost, shortage cost, holding cost, etc. are predetermined. However, there will be some variations. As a result, fuzzy factors are more accurate to deal with the real world?s problems. This research attempts to cut down the cost in a fuzzy environment by using quadratic demand, shortage, Weibull deterioration rate, and variable holding cost. Costs such as ordering, shortage, and deterioration are addressed as pentagonal fuzzy numbers that are defuzzified using a graded mean representation approach. Finally, sensitivity analysis was carried out to investigate the influence of cost parameters on total inventory cost. A numerical example is used to validate the proposed model in a real-world system.

Optimal pricing and advertisement with allowable shortages for non-instantaneous deteriorating items under inflation and trade credit

Optimal pricing and advertisement with allowable shortages for non-instantaneous deteriorating items under inflation and trade credit

A bi-objective non-linear approach for determining the ordering strategy for group B in ABC analysis inventory

The main aim of this research is to find the best inventory review policy for different types of items in group B in ABC analysis through minimizing the total cost of the system and maximizing the service level. Moreover, this study has considered several operational constraints such as limitations on storage space, number of orders, and allowable shortage. To solve this problem, first, an individual optimization method is utilized to obtain optimal solutions. Then, two classic and novel multi-objective optimization methods have been used to convert the bi-objective problem to a single-objective and reach the near-optimal solutions for both objectives simultaneously. Finally, the proposed methods are compared in terms of objective function values and computational time to find the better method.

A sustainable inventory model with controllable carbon emissions in green-warehouse farms

Global warming has become a sensitive issue, and many countries are trying to control CO2 emissions by investing in many projects and promoting green industrial practices. This study proposes a sustainable price-reliant demand inventory model with controllable carbon emissions to reduce the CO2 emissions from farm warehousing activity. This study extends previous research that has proposed a sustainable inventory system with greenhouse facilities by controlling carbon emissions through green investment. This study involves an investment in energy-efficient green technology in a two-warehouse inventory system to curb carbon emissions during the transportation of products from the owned to the rented warehouse and also to the final customers. Two warehouses are frequently used in practice to avoid stock-out situations and make the business more profitable. Case 1 represents the model with allowable shortages, and Case 2 represents the model without shortages. Pricing strategies and a hybrid payment scheme are applied throughout this study to make the business more profitable as well as to entice new customers. The retailer has to prepay some portion of the purchasing cost, with the payment scheme requiring multiple installments. Non-instantaneous deterioration is considered in this study to show the lifetime of product freshness in the presence of a constant backorder. A nonlinear model is proposed, and a solution algorithm is suggested. A real case study is introduced, which provides an illustration of the proposed model. The examples are solved and are followed by a sensitivity analysis that reveals the validity of the model, and conclusions are presented.

Stackelberg models in two-level supply chain with imperfect quality items with allowable shortages

Application of an absolute supply chain model does not invalidate the possibility of few defective items in a supplied lot, therefore it becomes essential to conduct an inspection process for segregating the defective items, subsequently such segregated items are sold at discounted price. Shortages mainly occur with sudden demand or erratic production capacity, and player’s decisions are influenced by it. In this paper, the shortage is considered as a seller’s decision variable and demand is receptive to selling price and marketing expenditure of the buyer. Player’s interaction will be reviewed and determined as non-cooperative Stackelberg game. Further, a supply chain model is endured to substantiate the interaction and democracy among buyer and seller in the supply chain and is pitched by non-cooperative game theoretical approaches. The Stackelberg game approach is used in the non-cooperative method where one player acts as leader and another as follower. Hereafter, unanimous numerical examples along with sensitivity analysis are exhibited to compare amidst two different models with and without shortages to demonstrate the significance of the paper.

Effect of warranty and quantity discounts on a deteriorating production system with a Markovian production process and allowable shortages

This paper explores the retailer's optimal lot sizing and quantity backordering for a deteriorating production system with a two-state Markov production process in which quantity discounts are provided by the supplier. The products are sold with the policy of free reasonable repair warranty employing the fraction of nonconforming items in a lot size. Unlike the traditional economic production quantity (EPQ) model with warranty policy based on the elapsed time of the system in the control state follows an exponential distribution, this paper not only constructs an alternative mathematical model for EPQ model based on the fraction of nonconforming items in a lot size for an imperfect production system but also extends the topics of optimal quantity and shortage to a wider scope of academic research and further finds that some results are different from the traditional EPQ models. We seek to minimize the expected total relevant cost through optimal lot sizing and quantity backordering. We also demonstrate that the optimal lot size is bounded in a finite interval. An efficient algorithm is developed to determine the optimal solution. Moreover, a numerical example is given and sensitivity analysis is conducted to highlight management insights.

Fuzzy inventory model with allowable shortages and backorder

Fuzzy inventory model with allowable shortages and backorder

Replenishment policy for deteriorating items with trade credit and allowable shortages under inflationary environment

In this paper we have developed an inventory model for deteriorating items in which demand of the product is a function of expiration date. In such cases the demand for the product generally decreases as the product is nearer to its expiration date. The different cases based on the allowed trade credit period have been discussed under inflationary environment. Shortages are allowed and partially backlogged. To exemplify this model numerical examples for all the cases and sensitivity analysis with respect to important parameters have been provided. The convexities graphs and the variation in total cost with the changes in parameters are represented with the help of the graph. The objective of this paper is to find out the optimal ordering quantity by which one can minimise the total associated cost of the system.

Replenishment policy for deteriorating items with trade credit and allowable shortages under inflationary environment

In this paper we have developed an inventory model for deteriorating items in which demand of the product is a function of expiration date. In such cases the demand for the product generally decreases as the product is nearer to its expiration date. The different cases based on the allowed trade credit period have been discussed under inflationary environment. Shortages are allowed and partially backlogged. To exemplify this model numerical examples for all the cases and sensitivity analysis with respect to important parameters have been provided. The convexities graphs and the variation in total cost with the changes in parameters are represented with the help of the graph. The objective of this paper is to find out the optimal ordering quantity by which one can minimise the total associated cost of the system.

Economic manufacturing quantity model with machine breakdown and deteriorating production process

In this study, the author proposes a jointly setting model for determining the process mean and economic manufacturing quantity (EMQ) under the machine breakdown and deteriorating production process. The system addresses the corrective maintenance, preventive maintenance, and allowable shortage. The quality loss of conforming product is considered and Taguchi’s asymmetric quadratic quality loss function is used for evaluating the product quality. The optimal process mean and economic manufacturing quantity are jointly determined by minimizing the total expected cost of product per unit time including the set-up cost, holding cost, corrective maintenance cost, preventive maintenance cost, shortage cost. A solution procedure is devised to obtain the optimal solution and the sensitivity analysis of key parameters is conducted to investigate the effect on the optimal solution.

Selling Price Dependent Demand with Allowable Shortages Model Under Partially Backlogged—Deteriorating Items

In this paper an inventory model for items with linear deterioration rate is studied. Demand depends upon the selling price of the commodity i.e. if selling price increases, the market demand decreases and vice versa. Shortages allowed and partially backlogged. There is neither repair nor replacement of deteriorated units occurring during the cycle. The ordering cost, purchasing cost, deteriorating cost, holding cost and the lost sale cost are considered in the inventory management. The model is solved analytically by minimizing the total inventory cost. Finally results are analyzed and demonstrated with illustrative numerical example. Sensitivity analysis is carried out to study the changing of different system parameters of this model.

A fuzzy inventory model for linear deteriorating items with selling price dependent demand and allowable shortages under partially backlogged condition

In the present study, we investigate a fuzzy inventory model for linear deteriorating items in which demand depended upon the selling price. Shortages are allowed and are partially backlogged with a rate dependent on the duration of waiting time up to the arrival of next lot. This study has been done with two types of model; first one is crisp model and second is fuzzy model. In crisp model the parameters are considered as real numbers and in fuzzy model, these are taken as triangular fuzzy numbers. Graded mean integration representation method, signed distance method and centroid method are used to defuzzify the total cost function. Numerical example is given to validate the model and sensitivity analysis of different system parameters are carried out to study the effect of changes by using Mathematica software.

Economic order quantity in fuzzy sense with allowable shortage: A Karush Kuhn-tucker conditions approach

Economic order quantity in fuzzy sense with allowable shortage: A Karush Kuhn-tucker conditions approach

Retailer’s optimal ordering policy for deteriorating items under order-size dependent trade credit and complete backlogging

The retail inventory management literature generally assumes that suppliers seek to stimulate demand by offering retailers a delay in payment. In practice, however, suppliers tend to offer retailers a partial delay in payment. To accommodate this possibility, this paper establishes an economic order quantity inventory model for deteriorating items, with allowable shortages and permissible partial delay in payment based on the order quantity. This paper presents theoretical results to determine the optimal replenishment time and the length of time for the stock to draw down completely, and with these time values the optimal ordering and backlogging policies are calculated for the retailer in order to minimize the total inventory cost per unit time. The optimal solutions are obtained analytically. The inventory model is validated numerically. A sensitivity analysis of the optimal solution with respect to the parameters of the inventory system and managerial insights are given. The proposed inventory model reduces to some existing inventory models.