A Fuzzy Random Continuous (QrL) Inventory Model Involving Controllable Back-order Rate and Variable Lead-Time with Imprecise Chance Constraint

  • Debjani ChakrabortyEmail author
  • Sushil Kumar Bhuiya
  • Debdas Ghosh
Conference paper
Part of the Springer Proceedings in Mathematics & Statistics book series (PROMS, volume 253)


In this article, we analyze a fuzzy random continuous review inventory system with the mixture of back-orders and lost sales, where the annual demand is treated as a fuzzy random variable. The study under consideration assumes that the lead-time is a control variable and the lead-time crashing cost is being introduced as a negative exponential function of the lead-time. In a realistic situation, the back-order rate is dependent on the lead-time. Significantly large lead-times might lead to stock-out periods being longer. As a result, many customers may not be prepared to wait for back-orders. Instead of constant back-order rate, we introduce the back-order rate as a decision variable, which is a function of the lead-time throughout the amount of shortage. Moreover, a budgetary constraint is imposed on the model in the form of an imprecise chance constraint to capture the possible way of measuring the imprecisely defined uncertain information of the budget constraint. We develop a methodology to determine the optimum order quantity, reorder point, lead-time, and back-order rate such that the total cost is minimized in the fuzzy sense. Finally, a numerical example is presented to illustrate the proposed methodology.


Inventory Imprecise chance constraint Fuzzy random variable Possibilistic mean value 


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Copyright information

© Springer Nature Singapore Pte Ltd. 2018

Authors and Affiliations

  • Debjani Chakraborty
    • 1
    Email author
  • Sushil Kumar Bhuiya
    • 1
  • Debdas Ghosh
    • 2
  1. 1.Department of MathematicsIndian Institute of Technology KharagpurKharagpurIndia
  2. 2.Department of Mathematical SciencesIndian Institute of Technology (BHU) VaranasiVaranasiIndia

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