Title: Determination of advertisement control policy for complementary and substitute items for a class inventory problem

Authors: D. Khatua; K. Maity; S. Kar

Addresses: Department of Basic Science and Humanities, Global Institute of Science and Technology, Haldia, Purba Medinipur-721 657, West Bengal, India; Department of Mathematics, National Institute of Technology, Durgapur-713209, West Bengal, India ' Department of Mathematics, Mugberia Gangadhar Mahavidyalaya, Bhupatinagar, Purba Medinipur-721 425, West Bengal, India ' Department of Mathematics, National Institute of Technology, Durgapur-713209, West Bengal, India

Abstract: The optimal advertisement for complementary and substitute items are formulated to control the demand on defective production assortment inventory system. Here, the advertisement rate is unknown and considered as a control variable. The demand is controlled by stock as well as the advertisement rates. The production rate is stock dependent and vanishes when the stock reaches to its maximum value (which is also called carrying capacity for stock). The rate of defective production is constant and the inventory control problem is developed for infinite time horizon. The asymptotically stable point is also analysed by negative eigenvalue, which is shown graphically. The total profit of the system consisting of the sales proceeds, production cost, inventory holding cost and advertisement cost is formulated as a Pontryagin's optimal control problem for steady state. The effect of discounting/time value of money is also taken into consideration. The model is simulated numerically for two items and the optimum results are presented both in tabular form and graphically. Our results show that a joint optimisation of the inventory and advertisement policy leads to a better system performance and increase the profit.

Keywords: inventory problem; optimal advertisement policy; stock dependent production rate; complementary and substitute items; asymptotical stability analysis; infinite time horizon.

DOI: 10.1504/IJBFMI.2017.085363

International Journal of Business Forecasting and Marketing Intelligence, 2017 Vol.3 No.3, pp.223 - 247

Received: 27 Sep 2016
Accepted: 15 Jan 2017
Published online: 24 Jul 2017 *

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