# Optimal Strategies for Deteriorating Inventory Systems Under Trapezoidal Type Demand

Kunal Tarunkumar Shukla (Lukhdhirji Engineering College, India) and Mihir S. Suthar (Charotar University of Science and Technology, India)
DOI: 10.4018/978-1-5225-3232-3.ch001
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## Abstract

In this chapter, we study different inventory systems with trapezoidal demand rate, i.e., demand rate is a piecewise linear and continuous function. This chapter presents mathematical formulations of optimal replenishment policies for items with trapezoidal demand rate. Section 1 presents detailed literature survey for inventory systems with ramp type and trapezoidal type demand. In Section 2, Formulation technique for inventory system of items, which follows trapezoidal type demand rate. Section 3 presents effect of deterioration in model discussed in Section 2. Optimal strategy for deteriorating items with expiration dates under trapezoidal type demand and partial backlogging is discussed in Section 4. In Section 5, sensitivity analysis is carried out and chapter is concluded along with future research scope in Section 6.
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## 1. Introduction

Modeling inventory systems deals with determining the level of goods or services; that is required to maintain for smooth business operations. The idea for the decision is a model that balances the cost of capital resulting from holding too much inventory against the the penalty cost due to inventory storage. The prime factor affecting the solution is the nature of demand. Demand may be deterministic or probabilistic. Demand of items which are being used regular basis; daily or monthly is almost constant, like milk, grains, household electricity consumption etc. Otherwise, it may be depending upon time, price and quantity. Moreover, demand changes as per need of the product in market. It is very difficult to develop a general model that covers all possible situations. This chapter includes representative models for different situations with trapezoidal type demand rate. When you study the different models, you will notice that the solution uses different algorithms, including calculus, linear, non-linear programming etc. Regardless of the tool used to solve the model, you should always keep in mind that any inventory model sees two basic results: How much and when to order.

The trapezoidal type demand rate is an extension of ramp type demand rate. The inventory model with ramp type demand rate was proposed the first time by Hill (1995). Hill considered increasing demand followed by constant demand, and derived the exact solution. Mandal and Pal (1998) extended the inventory model with ramp type demand for deteriorating items and allowed shortages. Wu and Ouyang (2000) extended the inventory model to include inventory policies, like, models starting with no shortages and models starting with shortages. Deng, Lin and Chu (2007) pointed out some results derived by Mandal and Pal (1998) and Wu and Ouyang (2000), and resolved the same problem by thorough and efficient solution to derive an optimal solution. Further, more development was done for inventory models with ramp type demand, by Wu (2001), Giri et. al (2003) etc.

In case of fashionable or seasonal goods, it is observed that inventory system consists of several replenishments and each order cycle is of fixed length. The rate of demand of such items increases with respect to time; up to certain time and then it becomes stable and remains constant for some time, and thereafter it starts decreasing to zero or a constant, and then next replenishment starts. Such demand pattern is defined as trapezoidal type demand. For such items, Cheng and Weng (2009) extended inventory model with ramp type demand to trapezoidal type demand rate. Chung (2012) found some shortcomings and removed them by providing the rigorous methodology to avail an optimal solution. One may refer, Cheng, Zhang and Wang (2011) developed inventory model for time dependent deteriorating items with trapezoidal type demand rate and partial backlogging rate which is non-increasing exponential function of the waiting time up to the next replenishment. Taleizadeh and Nematollahi (2014) derived EOQ model with constants deteriorating rate and demand under backordering and delay payment for a perishable item over the finite horizon planning. Lin, Hung and Julian (2014) studied optimal policy for deteriorating items with trapezoidal type demand and partial backlogging by Cheng et al. (2011) and created a note for some questionable results in the models and their solutions. The inventory models for deteriorating items with trapezoidal type demand with assumptions of shortages and without shortages have been developed by Wu, Skouri, Teng and Hu (2015) to maximize the net present value of total profit included the purchase costs and time varying deteriorating rate. More relevant citations are Bakker et. al (2012), Chen & Teng (2013), Deng et. al (2007), Shah et.al (2014), Shukla et.al (2016), Wang et.al (2014) and their references.

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