OCTOBER 19-20, 2012 - YMCA University of Science & Technology

Proceedings of the National Conference on
Trends and Advances in Mechanical Engineering,
YMCA University of Science & Technology, Faridabad, Haryana, Oct 19-20, 2012
OPTIMIZATION OF INVENTORY MODEL FOR DECAYING ITEM
WITH VARIABLE HOLDING COST AND POWER DEMAND
Ankit Prakash Tyagi 1 , Rama Kant Pandey 2 , Shivraj Singh 3
1 Dayanand Brijendra Swarup (PG) College, Dehradun, Uttarakhand, India
2 Department of mathematics, Dayanand Brijendra Swarup (PG) College, Dehradun, Uttarakhand, India
3
Department of mathematics, D.N. (PG) College, Meerut (UP), India
e-mail: ankitprakashtyagi88@gmail.com
Abstract
In inventory control phenomena, holding cost is an integral part of total cost of every inventory system. This is
determined from the investment in physical stocks and storage facilities for items during a cycle. In most of the
inventory research papers with power demand pattern, holding cost rate per unit time for perishable inventory is
assumed as constant. However, this is not necessarily the case when items in stock are decaying. In the present
work, paying better attention on the holding cost, we present a deteriorating inventory model in which the unit
holding cost is based on the deterioration of the inventory with the time the item is in stock. The deterioration is
assumed Weibull distributive. The power pattern of demand is considered in this paper. Shortages are allowed
and partial backlogged. The partial backlogging rate is a continuous inverse function of waiting time in
purchasing the item during stock out period. By using classical optimization technique, conditions for uniquely
existence of global minimum value of the average total cost per unit time are discussed. Numerical illustration
and sensitivity analysis are presented.
Keywords: Inventory, Deterioration, Shortage, Weibull Distribution, Power demand, variable holding cost.
1 Introduction
Food items, drugs, pharmaceuticals and agricultural products are few examples of essentially required items to
fulfill our daily requirements. In practice, appreciable deterioration can take place during the normal storage
period of such items and consequently this loss must be taken into account when making plan of our
consumption. Therefore, in many inventory models, the effect of deterioration is very important assumption. To
the best of our knowledge, an EOQ model for an inventory with variable proportion of the on-hand inventory
deteriorates with time have first been developed by Ghare and Schrader [1]. Later, many prominent researcher
papers on inventory control have been considered by Covert and Philip [2], Misra [3], Shah [4] etc. In their
investigations, the market demand for the item is considered to be constant. As time progressed, several other
researchers developed inventory models with decaying items and time dependent demand rates. In this
connection, works done by Mandal and Pal [5], Wu et al. [6], Teng et al. [7], Rau et al. [8], etc. are noteworthy.
Datta and Pal [9] investigated an inventory system with power demand pattern for items with variable rate of
deterioration.
In some situations of inventory control, demand before ending season exists and the inventory has mostly
consumed through joint effect of the demand and the deterioration. This type of situations laid the foundation of
stock out phenomena. Consequently, when stock out state occurs, some customers are willing to wait for
backorder and others would turn to buy from other sellers. Many researchers such as Park [10], Hollier and Mak
[11] and Wee [12] considered the constant partial backlogging rates during the shortage period in their inventory
models. In most of inventory systems, the length of the waiting time for the next replenishment would decide
whether the backlogging will be accepted or not. Therefore, the backlogging rate is variable and dependent on
the waiting time for the next replenishment. Chang and Dye [13] investigated an EOQ model allowing shortage
and partial backlogging. They assumed in their inventory model that the backlogging rate is variable and
dependent on the length of the waiting time for the next replenishment. Many researchers have modified
inventory policies by considering the ‘‘time-proportional partial backlogging rate’’ such as Abad [14],
Papachristos and Skouri [15], Wang [16], Papachristos and Skouri [17], Yang et al. [18] etc.
In the cited inventory researcher papers above, holding cost per unit time is taken as constant. Inventory decision
maker generally has to face the major concern to hold the decaying items. For smooth running of business
inventory should be available for whole cycle time in good condition. So inventory holding cost is an integral
term of total cost function of inventory and for better demonstrating the real life situations, holding cost per unit
should be more general as possible. Weiss [19] noted that variable holding costs are appropriate when the value
of an item decreases the longer it is in stock; Ferguson et al. [20] recently indicated that this type of model is
suitable for perishable items in which price markdowns or removal of aging product are necessary. Goh [21] first
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