 # Application of Simulation Techniques: Reducing the “Givens” and “Approximations” in the Analysis of Stochastic Inventory Models

Ningombam Sanjib Meitei (Devi Ahilya University, India) and Snigdha Banerjee (Devi Ahilya University, India)
DOI: 10.4018/978-1-4666-9888-8.ch014
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## Abstract

In the present work, we provide a simulated inventory model incorporating multiple stochastic factors affecting an inventory model. This can provide solutions to managerial problems faced by retailers that have been addressed through the Single period problem (SPP) models. For a time dependent SPP with multiple discounts of random amounts at random time points, we consider a model wherein the factors demand rate, lead-time, number of discounts during a season, discount rates, time epoch at which a new discount rate is offered are stochastic. We provide solution procedures as pseudo algorithms for simulating near optimal order quantity and estimate of rate of price decline as well as optimal values of order quantity and total expected profit for a given value of initial selling price. Illustrative examples are presented in order to enable the researchers to be able to apply the methodology explained. The technique for estimating the probability that a business system shall be profitable or be a loss venture is demonstrated using numerical example.
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## Introduction

“Inventory” may be defined as the stock that consists of raw materials, goods in process or finished goods maintained by a business or industry so as to ensure smooth operation. This involves optimal determination of factors such as

• 1.

The order quantity,

• 2.

Time of placing the order,

• 3.

Selling price,

• 4.

Discount rate, etc.

If some of the factors influencing the inventory model are known only in terms of their probability distributions, then such models are called stochastic inventory models. Newsboy Problem or Classical Newsboy Model (Arrow et al., 1951) is perhaps the most celebrated model in operations management. The objective of this model is to find the optimal order quantity in a stochastic demand framework with single procurement opportunity at the beginning of the selling season with deterministic season length. Unsold items are either sold out at a discounted price (salvage value) or disposed off for a cost i.e., at overage cost. In case of stock-outs, the retailer faces lost sales. The inventory models that are developed using the extended/generalized newsboy model settings are broadly categorized as single period problems (SPP). In this work, we consider a SPP model.

A realistic inventory model may involve multiple random factors. Even though considering these random factors will render the model closely reflective of real life situations, the analysis of such models becomes increasingly complex. For such models, analytical investigation of the functional forms of total cost, profit, revenue etc. may be almost incomprehensible and hence managerial insights are difficult to obtain. This has brought in bottlenecks in consideration of realistic stochastic models in inventory literature and resulted in most of the work assuming a deterministic platform. As observed in the extant literature, in order to obtain tangible solutions from inventory models with multiple stochastic factors, following tactics have been employed by researchers:

• 1.

Ignore Some of the Factors: Many researchers made the assumption of no stock-out shortage or zero penalty cost. For example, the newsboy model presented and analyzed by Granot and Yin (2005), did not consider penalty cost. They pointed out that when penalty cost (or goodwill cost) is considered, no closed form expression for equilibrium decisions and profit functions are available for any newsboy model that considers price dependent stochastic demand functions. Similarly, Lau and Lau (1988) also presented a stochastic inventory model with price dependent demand where price can be decreased in order to increase the demand and the random component of the additive demand model follows normal distribution. They obtained closed form solutions for order quantity and price only when the penalty cost is assumed to be zero.

• 2.

Assume Parameter Values to Be Given: Assuming that some of the parameter values are known. For example, Khouja (1995) investigated a newsboy problem with progressive discounts wherein he considered discount rates as given parameter values. Petruzzi and Dada (1999) considered the parameters of the additive demand models to be known. For a given demand-price relationship and a given order quantity, Khouja (2000) obtained optimal discounting policy for a given realization of demand under the assumption that the discounts are equally spaced in terms of price on the domain [0, Maximum Selling Price]. We denote this maximum selling price by S0.

• 3.

Approximate Random Factors by a Deterministic Function: This is the more appropriate way of incorporating random factors into inventory models. The functional forms of the factors may be investigated using empirical data and parameter values may be estimated e.g., price pathway during a selling season has been approximated by a linearly declining function in Banerjee and Meitei (2010), Meitei and Banerjee (2012, 2013).

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