This article deals with the novel approach of solving a village supply chain model (hut inventory) studied in all rural areas. First of all, we develop a crisp mathematical model considering a parabolic demand function in the entire inventory process during the day time of approximately 10 h. At the peak of the business hours, the demand of the commodities becomes high and then it began to fall down and finally becomes zero, if some of the excess amount exist after the end of the day then it would be used for lost sale/exhaust under special rebate. The objective of the research is to find the actual optimal order quantity, optimal unit selling price and the optimal cycle time so as to maximise the overall system profit as well as the reduction of system cost. The unit selling price is set through the negotiation of the purchaser and the seller followed by a bargaining function. The demand rate of the customers and the purchasing price of the selling items by the retailer (seller) per unit item are assumed to the fuzzy stochastic variables. To solve the model, the concept of duality has been developed based on alpha-cuts and its dual beta-cuts of fuzzy numbers. The numerical result shows that the discount model has the capability to achieve the higher per capita profit return 58.42% (cycle time 6.76 h, order quantity 251.19 kg) for lower investment of the retailer, but it attains maximum 65.94 % (cycle time 7.95 h, order quantity 312.96 kg) per capita profit return with respect to the all-items exhaust and that for partial item's sale the per capita profit return reaches to 43.30 % (cycle time 10.50 h, order quantity 431.46 kg) all the time. However, the comparative analysis and graphical illustrations have been discussed to show the actual novelty of the proposed approach.

This article deals with an application of fuzzy logic on determining the talent identification of pre-pubertal, pubertal and post-pubertal soccer players based on their body composition, physical fitness and physiological variables. A total of 90 male volunteers (age: 10–16 years) regularly playing soccer have been included from a football camp held at Salboni, Midnapore District, West Bengal, India. They have been divided into three age groups (i) pre-pubertal (n = 30), (ii) pubertal (n = 30) and (iii) post-pubertal (n = 30). The volunteers have been acclimatized for 15 days and selected body composition, physical fitness and physiological variables are measured. Data have been statistically treated with one-way ANOVA followed by multiple-comparison tests and fuzzy logic systems. Results showed that ANOVA predicts which age group showed better values for each selected variables but it fails to predict the ideal value of body composition, physical fitness and physiological variables to be considered in talent identification among each age group because of the variability of the data under different situations. To validate the study, a fuzzy logic system was applied to get a clear motivation over the subject. The fuzzy logic system accurately predicts the variables for talent identification. The fuzzy system gives the highest satisfaction levels beyond the ANOVA test and hence it may be applied to the determination of sports talent identification.

This article develops a fuzzy portfolio economic production quantity (EPQ) model with the effect of the time value of money. First of all, we construct a nonlinear crisp bi-objective (profit maximization and inflation minimization) mathematical problem with the effect of inflation over time on the unit selling price and all cost components explicitly. Initially, the bi-objective problem is solved with the help of the global criterion method. However, due to the uncertain nature of the demand rate and the production rate, they are assumed to be triangular fuzzy numbers. Then we construct two nonlinear power membership functions of portfolio return and portfolio risk with respect to the profit and cost deviation functions to form a fuzzy portfolio model of a single-objective function of profit maximization problem under the optimum cycle time, optimum production run time, portfolio risk and return variables subject to some constraints. The fuzzy problems are solved to achieve the highest aspiration level and some other design variables. A case study has been analyzed for numerical illustrations and comparative analysis to validate the model. Comparative analysis reveals that the fuzzy portfolio approach gives + 21.78

This study presents a game-theoretic analysis of a bi-level supply chain comprising a single vendor and multiple buyers, based on an integrated stochastic inventory model. The vendor offers trade credit to buyers who order a quantity above a certain threshold, encouraging larger orders and boosting overall supply chain performance. The production process is imperfect, leading to a proportion of defective items that are identified and returned to the vendor after a screening process at the buyer’s end. Both the defective rate and system lead time are treated as stochastic variables. The market demand is modeled as price-sensitive, influencing buyers’ ordering decisions. A key feature of the model is the allowance of planned shortages with full backlogging, assuming customers are willing to wait for the next delivery. The research contrasts decentralized and centralized decision-making strategies within the supply chain using a game-theoretic framework, capturing strategic interactions between the vendor and buyers. To solve the models, a stochastic genetic algorithm (GA) is employed, supported by numerical examples. The results reveal that centralized coordination significantly enhances the expected total profit of the supply chain compared to decentralized strategies. Sensitivity analyses further illustrate the impact of critical parameters such as demand rate, customer price sensitivity, defective product percentage, backorder cost, vendor’s interest loss, and buyer’s interest charges. The findings underscore the importance of supply chain integration and game-theoretic coordination, showing that planned backorders, despite associated costs, can lead to increased profitability when managed effectively. Graphical illustrations support the theoretical insights, and practical implications along with avenues for future research are discussed to validate and extend the model’s applicability.

The modern supply chain faces multiple challenges such as growing environmental concerns and increasing product quality, particularly in industries which dela with deteriorating items. This study introduces a novel framework that incorporates several carbon emission policies into an imperfect production system aimed at reducing deterioration and carbon emissions for a single vendor multiple-buyers supply chain. To enhance the storage conditions on product quality and environmental sustainability, two different kinds of technologies are incorporated: low-carbon and preservation. This study aims to optimize inventory decisions regarding pricing, different types of investments, production rate, inventory cycle time, and the number of production batches in order to maximize supply chain profit and simultaneously reduce emissions. Based on different carbon emission policies four different models are formulated and these problems are solved with the help of different solution algorithms. Due to the nonlinear nature of the formulated models, a metaheuristic optimization technique is incorporated alongside the traditional approach. For better comparative analysis, a numerical example is demonstrated with the help of two different approaches. It can be concluded that the goat search algorithm (GSA) performs better than the traditional approach (TA) for both economically and environmentally. The result shows that the model under without carbon policy, carbon tax, carbon cap-and-trade, and carbon cap-and-offset policy obtained respectively, 6

This article aims to develop a fuzzy bi-objective economic production quantity (EPQ) model where total demand splits in two parts, one for the online customers and the other for the offline customers. Here online demand is fraud-risk sensitive and follows a normal distribution, but offline demand assumes an intuitionistic fuzzy number. First of all, a crisp bi-objective (producer's profit and customer's cost optimisation) production inventory model has been solved under cyber-physical system. Then, due to the existence of uncertainty in offline demand, it is considered as a triangular intuitionistic fuzzy number (TIFN), and hence an intuitionistic fuzzy bi-objective mathematical model has been constructed. For model validation, by assuming offline demand as a triangular fuzzy number (TFN), a triangular fuzzy bi-objective model is formed to compare the result with the TIFN model. However, some comparative analyses have been discussed to show the validity of the model. All the models are solved by an entropy-based optimisation technique. To defuzzify the intuitionistic fuzzy model, the defuzzification method based on score and accuracy functions is utilised, and for the triangular fuzzy model, defuzzification is done by traditional approach. To show the practical usefulness of the proposed method, we have considered a real case study (picked up from a daily newspaper) for numerical illustrations. Numerical illustrations disclose that our approach is more profitable for decision maker, which gives +1.94% more profit than that of the existing one. Finally, sensitivity analysis and graphical illustrations also have been done to validate the proposed approach.

This article deals with an economic order quantity (EOQ) inventory model with varying numbers of customers and real-time dependent demand under a neutrosophic environment. First of all, we consider a cost minimization classical EOQ model and then solve it by using the calculus approach. But in practice, there exists a competitive atmosphere where some parameters of the model are assumed to be flexible in nature, and they follow the three-valued logic like a neutrosophic set. Thus, considering neutrosophic model, we extend this problem into a matrix game problem. To solve the model, we utilize a new de-neutrosophication method via the max–min approach of the matrix game, followed by a net aggregated score of neutrosophic elements alone. A new solution algorithm has also been developed for numerical computation over a case study dataset. A comparative numerical analysis has been done to show the novelty of the proposed approach under the recent five existing methods on neutrosophic decision-making models. Our findings reveal that the inventory system cost differs significantly with respect to the existing state-of-arts. However, the standard score (z-score) statistical interpretations show, for the cases of the max–min matrix game with aggregation operator, the mean and standard deviation of inventory system cost are assumed to be 1240.64 and81.78, respectively. But it becomes 1327.51 and59.97 for the cases of other methods. The corresponding z-scores are − 1.975 and − 1.245 about the primal (initial) solution. The z-score of the numerical outputs obtained by the new method shows the novelty and hence validates the proposed approach. Finally, the managerial insights, advantages, limitations, and a conclusion have been incorporated, followed by a scope of future work.

In this article, a game problem involving the traditional backlogged Economic Production Quantity (EPQ) model with each player's lock fuzzy strategic behaviour is discussed. Due to the model parameters' flexibility, fuzzifying the model we develop a fuzzy game problem which can be split into the problems of four incomplete games according to the nature of strategy/key vectors like (in)dependent, partially dependent and the case of neither dependent nor independent (skew strategy) over the cost objective function. A new methodology has been developed for defuzzification of the proposed fuzzy model. Numerical illustrations along with comparative analysis have been done via case study using the LINGO software where the keys of both the players (Consumer and Producer) are one of the vital decision variables. Our findings show that assuming indifferent attitudes from both players reduces the inventory system cost. The novelty of the proposed approach has been justified by the sensitivity analysis, graphical demonstrations and conclusion with managerial insights.

This study discusses a model of an imperfect production inventory system with deterioration, controllable emission of carbon and its reduction effort. The controllable service facility provided by the manufacturer to the customer along with preservation technology investment. The main objectives of this study are to determine optimal decision variables of the production process via maximization of annual total profit of the manufacturer. First, we find analytical forms of all the dynamic variables like production rate, carbon emission level, carbon emission reduction investment, controllable service level and service investment. To obtain the objective function with some constraints, we utilize Pontryagin's Maximum Principle over controllable dynamic variables. For numerical computations, few metaheuristic algorithms are employed to find the optimum strategies along with their sensitivity analysis. Considering standard normal data outputs under the 95% confidence interval, the optimum inventory profit lies above the mean profit of $3412.97 with a standard deviation of $123.56. Thus, it is suggested that the proposed Goat Search Algorithm always gives better optimum. We show that the investments in both service and emission reduction increase the profit up to 16%. This study has great applications because of its realistic assumptions and modelling from a case study on industrial problem.

This article deals with an industrial production process of a single item with safety stock and deterioration over time. First of all, we have considered an economic production quantity (EPQ) inventory model where the items are screened multiple times and the screening process itself has Type-I and Type-II errors. Some parts of the imperfect items are reworkable (serviceable) and the unusable items are discarded from the inventory instantly. Incorporating rework cost, disposal cost and screening cost in the inventory process, a total average system cost function has been studied and it has been optimized analytically. But to capture the flexibilities of the demand rate and all unit cost components (for comparative analysis) a fuzzy model has been developed. Indeed, we know that the defuzzification is a crucial step in any fuzzy inferential system, aimed at converting fuzzy outputs into equivalent crisp values for final decision-making. To get the model optimum we optimize the fuzzy membership function developed with the help of Newton's general interpolation formula for the proposed constrained non-linear optimization problem. The major novelties of this work include the construction of a new fuzzy membership function and techniques of decision making by means of a solution algorithm. For model validation, a numerical example has been analyzed on the basis of a case study and it has been compared with some of the existing methods. Findings reveal that the proposed method dominates others and up to 39.36% cost reduction is possible as a whole. Finally, sensitivity analysis and graphical illustrations have been carried out, followed by scope of future work.

Every company uses their best strategies to dominate others in the market and gets more profit. This study develops a bi-objective inventory problem where the several demand patterns have been analyzed under multiple fuzzy systems. First of all, we develop a crisp model through a case study of two laptop companies (I and II) considering the demand as a parabolic curve of time variable. The utility function method has been employed to solve the crisp model. After that, the model is converted to a fuzzy model by considering the total inventory cost of each company as triangular intuitionistic cloudy fuzzy number (TICFN). Various sub-models have been analyzed considering three cases and eight subcases depending on the intersection of two parabolic demand curves. A comparative study between crisp and other fuzzy models has been discussed here. The numerical study reveals that the decision-making with TICFN approach is more beneficial for any inventory practitioner. Because, the average profit gain is found to be 2.62

This article deals with a new solution procedure to solve a news vendor problem where the demand and cost parameters are assumed to be uncertain in nature. First of all, we develop a news vendor model where the total demand may or may not exceeds the order quantity. Basically, the demand function assumes a decreasing function over the unit selling price. Due to the market uncertainty (may be random or non-random) and nature of stock level with respect to the demand, the model itself has been split into two different sub-models and they are analysed under several uncertain environments. Considering all parameters random -fuzzy approximate reasoning, the model has been developed extensively. Initially, the model is solved in a traditional approach, then to find the best solution of the model, we further extend the model into non-linear stochastic programming problem and the problem of primal–dual model of fuzzy approximate reasoning via some evolutionary algorithms. For numerical computation and to validate the model, we consider a case study. Environmental cost is also computed to study the effect of total average system profit. Numerical study suggests that the dual cut approach provides the best optimum solution with respect to other methods whenever the demand gets value below the order quantity and the environmental cost varies with the items to be stocked. However, if the demand assumes greater than order quantity then the primal—dual cuts models give better results than other approaches. Indeed, sensitivity analysis and graphical illustrations are done for model justification. Finally, managerial insights, a concluding remark of the proposed study have been done followed by a scope of future work.

This article deals with the novel application of game theory utilizing fuzzy linguistic term set in a deteriorative production inventory model. First of all, a crisp deteriorative economic production quantity (EPQ) model is developed. Then considering the fuzzy flexibility of the cost and demand vectors of the model we construct a profit-maximizing fuzzy linguistic term matrix game problem under suitable constraints. To solve this problem, we utilize dominance rule to reduce a 2 x 2 matrix game through the application of similarity index for linguistic term fuzzy sets. However, to capture the uncertainty by means of fuzzy linguistic term sets De, S.K. invented a new fuzzy dichotomous index operator (DIO) named De's DIO (DDIO) associated with the reduced 2 x 2 fuzzy matrix game alone. The model is solved under several existing methods like traditional matrix dominance, compromise solution, and bi-objective optimization technique, etc., to validate the model via numerical simulation. It is also observed that for the fuzzy dominance (using DDIO) model, the average inventory profit is enhanced by on and from +28.54% to +89.89% with respect to the current state-of-the-arts. Finally, graphical discussions are made followed by a conclusion and scope of future work.

This article focuses on the development of metaheuristic algorithms based on the search process and collection of the oysters (living or non-living) from the sea bed in the day time. In the coastal belt, during tides, oysters are pushed by the water waves and gathered in specific places on the sea bed. For living oysters, they are generally found in the cold-water region, but for non-living shells, they can be found anywhere in the sea bed, depending on the blow of the wind and the path of the water wave. Since oysters have many economic values (because indoor decoration, ornaments and gift packs etc. are made using these sea shells), people from coastal zones usually collect them and plan for one of the best jobs/ trades in their social life. Inspiring from the nature of the collection procedure of the shells, we construct mathematical functions of the search path along which it can be available more. Utilizing this procedure into the field of optimization, we have found some noble results of a benchmark problem. Basically, we fit an appropriate test function for a benchmark problem, then some numeric experimental results are computed via oyster collection algorithms. A comparative study has also been made to validate the proposed approach. Moreover, considering the normality of the data, we have studied confidence intervals and analyzed statistically for the global acceptance of the method. It is also seen that the root mean square error (RMSE) for our proposed algorithms is 0.004007 and that for the other existing algorithms it becomes 0.007015. This gives the excellence success rate in the field of optimization with respect to others all the time. Indeed, the graphical illustrations show the efficacy of the proposed algorithms and, finally, a conclusion is made about keeping some scope for future works.

In this study, an Economic Production Quantity (EPQ) model with deterioration is developed where the production rate is stock dependent and the demand rate is unit selling price and stock dependent. The low unit selling price and more stocks correspond high demand but more stock corresponds to slow production because of the avoidance of unnecessary stocks. First of all, we develop the production model by solving some ordinary differential equations having deterministic profit function under some specific assumptions. Later, we develop the fuzzy model by solving the fuzzy differential equations using generalized Hukuhara (gH) derivative. In fact, the differential equation of the model has been split into two parts namely gH(L-R) and gH(R-L) on the basis of left (L) and right (R) alpha-cuts of fuzzy numbers for which the problem itself is transformed into multi-objective EPQ problem. A new formula of aggregation of several objective values obtained at different aspiration levels has been discussed to defuzzify the fuzzy multi-objective problems. We solve the crisp and fuzzy models using LINGO software. Numerical and graphical illustrations confirm that the model under gH derivative of (R-L) type contributes more profit which is one of the basic novelties of the proposed approach.

The modern global economy is becoming more challenging and it is hardly possible to minimize the inventory cost for inventory practitioners in the coming days. Basically, most of the enterprises deal with deteriorating items having flexible demand rate and follow natural idle time in the entire inventory process. Moreover, traditionally most of the research articles have been made under non-stop time frame, but in reality, in a day–night scenario there exists a natural idle time and hence the time consumed for inventory run time may be viewed as single shift or periodic model. Here we formulate an economic order quantity (EOQ) inventory model considering natural idle time and deterioration under some constraints and minimize the average inventory cost. Then, the model is converted into an equivalent fuzzy model, taking the demand and all the cost parameters as linguistic polynomial fuzzy set (LPFS). To defuzzify the model, we have adopted indexing method as well as α -cut method. To validate the novelty, numerical experimentations have also been analyzed with the help of metaheuristic and evolutionary algorithms like goat search algorithm (GSA) and particle swarm optimization (PSO). Comparative analysis reveals that GSA approach can give finer optimum (− 10

This article deals with a novel trade credit policy for growing items where the buyer collects the new born items from a supplier and sells them to the consumers when they are grown up. The supplier provides a credit period to the buyer through negotiation. From the buyer's view point first of all, a profit maximization growing items with carbon emission supply chain model has been studied under three different scenarios over the length of credit periods. To capture the non-random uncertainty of the model parameters and the learning effects we have assumed the demand rate of the consumers and all the cost parameters as triangular fuzzy linguistic terms set by means of a learning matrix. For defuzzification we have utilized the eigen values of the corresponding fuzzy learning matrices and solved the problem with the help of a solution algorithm under optimum number of new born items. However, for numerical computations we have utilized the data set studied by Alamri from a real case study. A comparative analysis has also been discussed for model validation. Finally, sensitivity analysis and graphical illustration are carried out to focus the novelty and to study the impact of the inventory parameters of the proposed approach.