The highly competitive and volatile nature of the current marketing environment makes it challenging to predict demand and other uncertain costs. To address this uncertainty, this study employs an interval-valued optimization technique. We propose an economic order quantity (EOQ) model within an interval framework, where the demand rate is represented as an interval-valued power function dependent on the green level, selling price, and time. Furthermore, the retailer’s purchasing and holding costs are treated as interval values; the purchasing cost is dependent on the green level, while the holding cost varies over time. The model also permits fully backlogged shortages, which are considered interval values. A parametric method is utilized to convert the differential equation for the inventory level from its interval form into a crisp equivalent. The resulting maximization problem is then solved using the Teaching-learning-based optimizer algorithm (TLBOA), after being converted to a crisp problem using interval order relations and the center-radius optimization approach. Finally, a numerical example is provided to illustrate and validate the proposed model, followed by a post-optimality analysis to examine the impact of various parameters on the optimal policy. (AMS classification code: 90B05, 49M37, 90C70, 90C59)

Advance payment for replenishment and the lifetime of an item are the most vital issues in the business sector. With the consideration of these issues, the formulation of a non-deterministic inventory model in the interval environment is of great importance. In this work, two different non-deterministic inventory models are formulated along with a variable holding cost, variable demand, and advertisement frequency under an advance payment policy in the interval environment. In the first inventory model, a no-shortage situation is discussed, whereas in the second model, a stock-out scenario is adopted by partially backordering during modelling. The modelling framework for both problems is accomplished based on interval differential equations. The interval optimization problems related to both models are obtained utilizing parametric approach and interval mathematics. Then, using interval order relations and different types of quantum particle swarm optimization, both interval optimization problems are solved. By means of two examples, the authentication of both models is examined and the benefits of the proposed approach are demonstrated. The impact of deviations of various parameters on optimality is studied graphically through the performance of sensitivity analyses, and the work is concluded by providing potential management insights and future scopes.

Tournamenting approach plays most vital and important role for selecting best individual in knockout tournament system. This idea is used for developing hybridization of algorithm based on teaching learning based optimization technique and named as tournament teaching learning based optimization algorithm (TTLBO). The main goal of this work is to apply hybrid TTLBO for solving maximization problems corresponding to the proposed non-instantaneous inventory model for single deteriorating item with trade-credit financing, partially backlogging and Weibull distributed deterioration. Demand depends on credit period and selling price of item. Now, our aim is to determine optimal order quantity, cycle length, selling price and maximum quantity of shortage by maximizing the retailer’s average profit. The validity of the developed model is tested with the help of an example. Also, the same example is solved by existing nine algorithms, viz. ABC, GQPSO, AQPSO, DE, RAO-1, RAO-2, RAO-3, HBO and TLBO algorithms to compare the performance and efficiency of the proposed TTLBO algorithm. Moreover, the analyses of sensitivity are studied to investigate the impact of different parameters involving in the model on the best found policy. Also, two nonparametric statistical tests, viz. Wilcoxon rank sum test and Friedman test are used to compare the statistical significant of proposed algorithm and existing nine algorithms. Finally, from numerical illustration and sensitivity analysis, a fruitful conclusion of this study is drawn.

Numerous studies have explored pricing and lot-sizing strategies for various payment methods, but most have focused primarily on the buyer’s perspective. This study, however, approaches these strategies from a different perspective, incorporating key and relevant factors often overlooked. The volume of sales increases when a seller accepts a buyer’s credit. However, it reduces sales volume when a seller requests a buyer make a payment in advance. To boost sales and profitability, a vendor occasionally provides a price reduction in exchange for a down payment. Demanding a down payment from a customer earns interest and carries without any risk of default. When a vendor offers customers the option to pay with credit, a higher delay payment period facility plan may boost sales volume, but it also increases the risk of default. To maximize profit per unit of time, the vendor aims to simultaneously determine the optimal selling price, replenishment schedule, and payment method. This is achieved by comparing and calculating the vendor’s profit per time unit for credit, cash, and advance payment options. This is done by comparing and calculating the seller’s profit for each piece of time for credit, cash, and advance payments. The following managerial impacts are highlighted by means of numerical analyses: (1) A particular payment type, among the three available options, yields the seller’s highest profit under certain conditions. (2) It is vitally crucial for a vendor to provide a price reduction if an advance payment is required. (3) Advance payment results in higher profit than delayed payment if sales volume does not significantly fall while switching from credit to advance payments, or vice versa. To solve the optimization problem, a popular metaheuristic algorithm (viz., Grey Wolf Optimizer) is used and finally performed a post optimality analysis for making a fruitful conclusion.

Objective: Rented house owners are take policy ensure their rent for a certain period for contract of rent. Due to pandemic situation, people have of less capital and are not possible to take high decision in business. Also, many of the food products are stored in high preservation, however when a food product collected out from preservative condition high rate of deterioration are seen among foods. Method: Lower capital feature of society force to consider an inventory system for three-parameter Weibull distributed deterioration, lees capital and borrowing of item from a greater business organization. Also, for security period of rent, new type of holding is considered in the proposed inventory model. Due to consideration of three-parameter Weibull distribution deterioration objective functions are highly nonlinear and to solve the optimization problems three types of quantum behaved particle swarm optimization technique applied. Findings: Gaussian behaved particle swarm optimization (GQPSO) technique gives better optimum than other techniques. A sensitivity analyses about some inventory parameters are shown in figures and fruitful managerial insight features are given for the model. Man power management in a business and duration of business period effects directly to profit. Novelty: Holding cost is constant for a period of time then linearly time dependent. This type of holding cost is a new concept in the literature. Discount on selling price is considered when items are started to deteriorated. Also, purchase cost paid at the end of business period and free from any interest charge. Keywords: Non­Instantaneous Deterioration, Stock Level, Trade Credit, Time­Varying Holding Cost, Low Finance, Weibull Distribution

It is commonly known that a number of variables, including price, supply levels, time, and green level, affect how quickly certain things are in demand. Furthermore, the inventory carrying cost is considered to be a nonlinear representation of time and is subject to variation throughout time. More precisely, it rises with time since longer storage times necessitate more costly warehouse space. This study presents a fully backlogged situation inventory system for a single commodity where the product’s selling price, green level, and time are used to simultaneously compute the demand rate in accordance with a power pattern. Purchase price is determined by the product’s nonlinear green level. Complete backorders are available for shortages. The impact of the product’s selling price, green level and time power function are combined to determine the product’s demand. Moreover, the holding cost also rises as the product is stored for a longer period of time. The primary goal is to determine the best inventory policy to maximise total profit per unit of time. Though the problem is highly nonlinear in nature. Hence, we cannot solve it analytically. To overcome these difficulties, we have applied several well-known popular metaheuristic algorithms (Water Cycle Algorithm (WCA), Artificial Electric Field Algorithm (AEFA), Teaching Learning Based Optimization Algorithm (TLBOA), Grey Wolf Optimizer Algorithm (GWOA), Sparrow Search Algorithm (SSA), Whale Optimizer Algorithm (WOA), Prairie Dog Optimization Algorithm (PDOA), Gazelle Optimization Algorithm (GOA), A Sinh Cosh Optimizer Algorithm (SCHOA) and White Sherk Optimizer Algorithm (WSOA), Archimedes Optimization Paradigm Algorithm (AOPA), Marine Predator Optimization Algorithm (MPOA), Geyser Inspired Algorithm (GIA), Runge Kutta Optimization Algorithm (RKOA), Lungs Performance-based Optimization Algorithm (LPOA) and Dwarf Mongoose Optimization Algorithm (DMOA)). It is observed that WCA perform better than other algorithms with respect to the convergence rate. A numerical example is taken in order to validate the proposed model. Finally, a post optimality analysis is performed in order to make a fruitful conclusion.

There has been a lot of research on pricing and lot-sizing practices for different payment methods; however, the majority has focused on the buyer’s perspective. While accepting buyers’ credit conditions positively impacts sales, requesting advance payments from purchasers tends to have a negative effect. Additionally, requiring a down payment has been found to generate interest revenue for the supplier without introducing default risk. However, extending the credit period, along with offering delayed payment options, has the potential to increase sales volume, albeit with an elevated risk of defaults. Taking these payment schemes into account, this study investigates and compares the per-unit profit for sellers across three distinct payment methods: advance payment, cash payment, and credit payment. The consumption rate of the product varies non-linearly not only with the time duration of different payment options but also with the price and the level of greenness of the product. The utmost objective of this work is to determine the optimal duration associated with payment schemes, selling price, green level, and replenishment period to maximize the seller’s profit. The Teaching Learning Based Optimization Algorithm (TLBOA) is applied to address and solve three numerical examples, each corresponding to a distinct scenario of the considered payment schemes. Sensitivity analyses confirm that the seller’s profit is markedly influenced by the environmental sustainability level of the product. Furthermore, the seller’s profitability is more significantly affected by the selling price index compared to the indices of the payment scheme duration and the green level in the demand structure.

In the current market situation, uncertainty in the market economy plays a significant role for proper controlling of inventory. In this situation, during an analysis of an inventory system with more realistic assumptions, it becomes a necessity to consider the imprecise behaviour of various inventory parameters. This work demonstrates a partially backlogged inventory model with interval uncertainty in which products during stock-in situation deteriorate at a certain rate. Two distinct cases are considered based on the advance payment and availability of discount. In the first case, the retailer makes full payment in advance to the corresponding supplier, and for this regard, supplier offers some price discount on the total purchasing cost. In the second case, the retailer makes partial payment in advance to the supplier but no discount facility is available. However, the retailer needs to make the rest of the payment at the time of receiving a lot from the supplier. As the inventory parameters are considered interval-valued, the objective function of the corresponding optimisation problem is converted into interval-valued, and to solve the same problem, the c-r optimisation technique is used. Two numerical examples are considered and solved using three distinct variants of QPSO to check the feasibility of the proposed model.

This study introduces an inventory system with a non-instantaneous deteriorating product with credit facility and variable demand depending on the selling price. Two different selling prices are considered in the deterioration and non-deterioration periods. Shortages are partially backlogged and dependent on the length of the customers’ waiting time upto the arrival of fresh lot. Alternative trade-credit policy is applied herein, and several cases, sub-cases and situations are investigated. The corresponding optimization problems of different cases, sub-cases and situations are solved using an interval-oriented multi-section technique with the help of interval mathematics and interval order relations. A numerical example with three different credit periods is studied and solved to validate the said problem. Also, two different case studies are investigated. Then to investigate the effect of changes of several system parameters on the optimal policy, post optimality analyses are performed.

Preservation of a product is an important issue in the inventory control system. It prevents the deterioration effect of the products while these are stored in the warehouse/showroom. Considering deterioration effect of the product and preservation technology, an inventory model of non-instantaneous deteriorating items is developed with the demand dependent on the selling price of the product. Two different preservation rates are considered. Shortages are allowed partially with two different backlogging rates. Due to consideration of three-parameter Weibull distributed deterioration and preservation facility, the corresponding optimization problems are highly nonlinear. So, these problems cannot be solved analytically due to nonlinearity. To overcome this situation, different variants of quantum-behaved particle swarm optimization (QPSO) are used. To illustrate and validate the proposed model, a numerical example is considered and solved for each case, and compared the results with the different variants of QPSO algorithms. Finally, a sensitivity analysis is performed to study the effect of changes of different parameters of the model on the optimal policy.

Generally, most of the inventory costs are not always fixed due to uncertainty of competitive market. In the existing literature, it is found that several researchers have worked on uncertainty considering inventory parameters as fuzzy valued. In this work, we have represented the inventory parameters as interval. Using this concept, we have developed a two-warehouse inventory model with advanced payment, partial backlogged shortages. Due to uncertainty, this problem cannot be solved by existing direct/indirect optimization technique. For this purpose, different variants of particle swarm optimization techniques (viz. PSO-CO, WQPSO and GQPSO) have been developed to solve the problem of the proposed inventory model by using interval arithmetic and interval order relations. Finally, to illustrate and also to validate the proposed model, a numerical example has been solved and the best found solutions (which is either optimal solution or near optimal solution) obtained from different variants of PSO have been compared. Then, a sensitivity analysis has been performed to study the effect of changes of different parameters of the model on the optimal policy.