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Inventory system with generalized triangular neutrosophic cost pattern incorporating maximum life-time-based deterioration and novel demand through PSO


At the manufacturing plant or while the products are being transferred from one supply layer to another, there is a considerable possibility of receiving damaged or faulty items mixed in with non-defective commodities. This research focuses on the non-defective and defective products that are shipped to retailers by their suppliers. The retailer reworks faulty items to make them non-defective, and the retailer receives a discount on the cost of purchasing defective items. The presented inventory system addresses the uncertainty in inventory costs and also considers the deterioration of items with prioritized maximum product life. In this study, our aim is to minimize the total inventory cost when demand rate as a function of quality and power pattern of time under crisp and generalized triangular neutrosophic environments. Based on the payment deal, interest charges are imposed only when the payment delay has passed a particular allowable time limit. The neutrosophic number, which provides three different types of membership functions representing truth, hesitation, and falseness, is used in the inventory model to handle the cost pattern’s uncertainty. A particle swarm optimization approach is used to analyze the proposed inventory model, and the results are validated using a numerical example and sensitivity analysis for various parameters.

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“The authors express their gratitude to the editors and referees for their encouragement and constructive comments on this article.”


This study did not receive any funding or Grants from any sources/company/agencies.

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Correspondence to G. Durga Bhavani.

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This inventory management study is developed using the following notation:


Ordering cost per order ($/order).


Selling price ($/unit).


Holding cost per item ($/unit/unit time).


Deterioration cost per item ($/unit/unit time).


Purchasing cost per unit item ($/unit).


Inspection cost per unit item ($/unit).


Rework cost per unit item ($/unit).


Rate of interest charged per year in stocks by suppliers.


Rate of interest earned by investment per year.

\(\theta (t)\):

Deterioration rate of items.


Maximum life-time in years of item, \((m>2)\).


Supplier permissible delay period.


Retailer product’s quality.


Cycle time in per cycle.


Non-defective rate of items \((0<f<1)\).


Supplier’s product quality \((0<r<1)\).

\(\gamma \):

Reduction percentage of purchasing cost of defective items.


Average demand per cycle \((d=\frac{x}{T}>0)\) .


Total demand per cycle.


Demand pattern index \((n>0)\).

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Bhavani, G.D., Mahapatra, G.S. Inventory system with generalized triangular neutrosophic cost pattern incorporating maximum life-time-based deterioration and novel demand through PSO. Soft Comput 27, 2385–2402 (2023).

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  • Deterioration
  • Quality
  • Triangular neutrosophic number
  • Delay in payment
  • Holding cost
  • Maximum life-time