Simultaneous pricing and inventory decisions for substitute and complementary items with nonlinear holding cost

  • M. A. Edalatpour
  • S. M. J. Mirzapour Al-e-HashemEmail author
Production Management


In practice, the holding cost in inventory models can be considered as linear or nonlinear functions. Nevertheless, nonlinear holding cost rarely attracted the attention of the researchers, and to our knowledge, it is novel in the presence of complementary or substitute items. It can also be interpreted as a way to include the perishability of products. Except for the perishability, nonlinear holding cost can be employed for other purposes like cold supply chain. In this paper, a new pricing method based on a multi-product inventory model is presented for complementary and substitute items. This paper aims to find the optimum value of the replenishment cycle and prices for the products supplied together (complementary) or interchangeable (substitute), such that the total cost of the inventory system is minimized. Demand is assumed as a price sensitive function while the proposed model is trying to obtain the optimal values of prices and the replenishment cycle simultaneously.


Pricing Complementary Substitute Nonlinear holding cost Multi-product Inventory 




Product index


Model type index

Decision variables


Order cycle


Demand of product i


Price of product i


Order quantity of product i


Revenue function


Item type



Fixed cost of ordering of product i


Total purchasing cost of product i


Average purchasing cost per ordering cycle (\(T\))


Base demand (demand for zero price)


Price sensitivity of the demand


Holding cost of product i per each unit of time


Purchase cost of product i


Ratio of complementarity between product \(i\) and its complement products \(\left( {0 \le \theta_{i} \le 1} \right)\)


Ratio of substitution between product \(i\) and its substitute products (\(0 \le \mu_{i} \le 1)\)


Deterioration rate


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Copyright information

© German Academic Society for Production Engineering (WGP) 2019

Authors and Affiliations

  • M. A. Edalatpour
    • 1
  • S. M. J. Mirzapour Al-e-Hashem
    • 1
    Email author
  1. 1.Department of Industrial Engineering and Management SystemsAmirkabir University of TechnologyTehranIran

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