Mathematical Problems in Engineering

Volume 2018, Article ID 6085342, 23 pages

https://doi.org/10.1155/2018/6085342

## Determining the Optimal Order Quantity with Compound Erlang Demand under (T,Q) Policy

^{1}SHU-UTS SILC Business School, Shanghai University, 20 Chengzhong Road, Jiading District, Shanghai 201899, China^{2}School of Mathematics and Statistics, UNSW, Sydney, NSW 2052, Australia

Correspondence should be addressed to Aiping Jiang; nc.ude.uhs@427pa

Received 10 March 2018; Revised 25 June 2018; Accepted 12 July 2018; Published 19 August 2018

Academic Editor: Neale R. Smith

Copyright © 2018 Aiping Jiang et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Abstract

Management of electric equipment has a direct impact on companies’ performance and profitability. Considering the critical role that electric power materials play in supporting maintenance operations and preventing equipment failure, it is essential to maintain an inventory to a reasonable level. However, a majority of these electric power materials exhibit an intermittent demand pattern characterized by random arrivals interspersed with time periods with no demand at all. These characteristics cause additional difficulty for companies in managing these electric power material inventories. In response to the above problem, this paper, based on the electric power material demand data of Shanghai Electric Power Company, develops a new method to determine the optimal order quantity in a cost-oriented periodic review system, whereby unsatisfied demands are backordered and demand follows a compound Erlang distribution. corresponds to the value of that gives the minimum expected total inventory holding and backordering cost. Subsequently, an empirical investigation is conducted to compare this method with the Newsvendor model. Results verify its superiority in cost savings. Ultimately, considering the complicated calculation and low efficiency of that algorithm, this paper proposes an approximation and a heuristic algorithm which have a higher level of utility in a real industrial context. The approximation algorithm simplifies the calculation process by reducing iterative times while the heuristic algorithm achieves it by generalizing the relationship between the optimal order quantity and mean demand interarrival rate .

#### 1. Introduction

When managing electric power material inventories, companies need to decide when to order and how much to order so as to maintain the appropriate inventory level. Considering that a majority of electric power materials exhibit an intermittent demand pattern, effective management of these electric power materials is essential to electric power companies. Intermittent demand appears at random, with time intervals during which no demand occurs. These characteristics make inventory control a rather challenging task. These electric power materials are usually characterized by low usage frequency, long intervals, and irregular demand. Furthermore, demand arrivals and sizes vary with usage and maintenance patterns, further adding to the degree of intermittence. Moreover, the historical data of electric power materials are limited, coupled with constraints on the forecasting method, demand forecast for these electric power materials is difficult. However, such electric power materials are usually essential components of expensive electric equipment. In the case of equipment failure, when electric power materials are unavailable and need to be ordered immediately, the cost of these urgent orders is higher than regular orders, significantly increasing the maintenance cost. In addition to this, the failure of a component in the electrical equipment is likely to lead to equipment failure, generating a high backordering cost and economic loss. On the other hand, excessive stock of electric power materials leads to backlogs and erosion, resulting in a significant waste of capital as well as the corresponding corporate cost. Some electric power materials stored in warehouses for a long time erode before they are put into use, while some are permanently idle due to the continuous changes and updates of electrical equipment. Since traditional inventory control is generally designed for regular demand, its application to intermittent demand is likely to lead to frequent stock-out events or excessive stock, making the inventory control of electric power materials a more challenging task. Thus, it is essential for electric power companies to control their intermittent electric power material inventories.

#### 2. Research Background

In this section, a brief review of the literature related to intermittent material inventory control is undertaken.

Traditional inventory control systems generally focus on regular demand and are not suitable for intermittent demand (Adelson [1]; Friend [2]). As a representative of traditional inventory control models, the Newsvendor model had been extensively used since it was introduced in 1962 (Hanssman [3]). This model states that if inventory is larger than demand, the remaining inventory will be sold at a discounted price or disposed at the end of period; however, if inventory is smaller than demand, some profit will be lost. Thus this model focuses on optimizing cost or profit and develops relevant inventory policy. Subsequently a great number of scholars make extensions to this traditional model which can be divided into three categories. Firstly, traditional models usually assume a fixed purchase price which departs from the reality. Thus price-sensitive demand is introduced. For instance, Xiao et al. [4] study the price-dependent demand in a multi-product Newsvendor model and propose corresponding methods of obtaining the optimal order quantity and optimal sales price. Kébé et al. [5] consider a two-echelon inventory lot-sizing problem with price-dependent demand and develop a mixed-integer programming formulation as well as a Lagrangian relaxation solution procedure for solving this problem. Secondly, considering the profound influence of demand variation on expected profits or costs, plenty of scholars focus on the modelling of demand when studying Newsvendor model. Uduman [6] employs demand distributions satisfying the SCBZ (clock back to zero) property to model the single demand for a single product, namely, newspaper, and subsequently obtain the optimal order quantity. Fathima et al. [7] study the generalization of Uduman’s model with several individual demands for a single product, followed by numerical illustrations. Furthermore, Faithma & Uduman [8] propose an approximation closed-form optimal solution for three cases of single period inventory model, in which the demand varies with the SCBZ property. In addition to SCBZ property, Kamburowski [9] developed a theoretical framework for analyzing the distribution-free Newsboy problems due to incomplete information. Similarly, Chen [10] considers unknown demand and demonstrates the appropriateness of applying the bootstrap method to Newsvendor model. Hnaien et al. [11] focus on discrete distributions of probabilities of demand. Besides, Priyan & Uthayakumar [12] consider a two-echelon multi-product multi-constraint inventory model with product returns and recovery. Lastly, two-level Newsvendor model including inventories on raw materials and finished products has recently received increasingly attention. In such model, Keramatpour et al. [13] develop a meta-heuristic algorithm to obtain the optimal solution which optimize profit and service level simultaneously.

Axsäter [14] reveals that a majority of inventory control models including Newsvendor model are built on some assumptions, for instance, that lead-time demand follows the normal distribution. However, they point out that although this assumption is reasonable in most cases, it is inconsistent with the reality in certain circumstances and leads to a waste of capital or low service levels. Thus, they propose the (*R, Q*) policy with the compound Poisson demand. Considering that the compound distribution model can determine arrival and demand sizes, respectively, and its structure is similar to the intermittent demand process, a vast number of scholars propose that compound distributions, compound Poisson distribution in particular, can offer a good fit for intermittent material inventory (Adelson [1]; Friend [2]). Similarly, Matheus and Gelder [15] also employ compound Poisson distribution to model demand and determine the order quantity and reorder points. Babai et al. [16] have developed a method to determine the optimal order-up-to level, subject to a compound Poisson distribution.

Dunsmuir and Snyder [17] present a method for determining an optimal reorder point for the continuous review (*S, Q*) inventory policy, subject to a compound Bernoulli demand. However, they fail to take the impact of lead-time demand on the service level into account. To fill the gap, Janssen et al. [18] make some modifications and adopt the (*R, s, Q*) inventory model to calculate the reorder point with a service-level restriction. Teunter et al. [19] restrict their study to the calculation of the mean and variance of lead time demand and derive closed-form expressions for the service level and the expected total cost under the (*T, S*) policy with a compound binomial demand, thereby avoiding complicated calculation of the mean and variance.

Of all the literature with compound demand distribution assumptions, the compound Poisson is the most widely utilized. Its popularity can be mainly attributed to the simplicity arising from the memoryless propensity of the exponential distribution of the demand interarrivals. Gupta et al. [20] introduce the Palm’s theorem to justify such demand modelling for whole complicated systems. However, in the case of one single material, this modelling is reported to lead to excess stock and imposes an additional financial burden on companies (Smith and Dekker [21]).The memoryless propensity implies a constant failure rate, which is inconsistent with the reality, where the likelihood of the failure of an item and the demand for it grows with time. Smith and Dekker [21] initially propose a more realistic renewal demand arrival process which has an Increasing Failure Rate (IFR), but they do not propose specific demand distribution to model the renewal demand arrival process. Subsequently Larsen and Kiesmuller [22] provide the first insight into the employment of compound Erlang distribution-k in the inventory control system and they derive a closed-form cost expression of the optimal reorder level for an (*R, s, nQ*) policy. Based on the same demand distribution, Larsen and Thorstenson [23] consider a continuous review base stock policy and demonstrate the use of a computer program to obtain the base stock level needed to achieve the specified order fill rate (OFR) and volume fill rate (VFR), respectively. Moreover, the respective performances of the OFR and VFR for different inventory control systems are compared (Larsen and Thorstenson [24]). With demand arrivals following the Erlang distribution, in conjunction with a Gamma distribution for demand size, Saidane et al. [25, 26] propose an algorithm to determine the optimal base stock level under a cost-oriented continuous review inventory system and conduct numerical investigations to compare the performance of this model in cost reduction with that of the compound Poisson distribution and the classic Newsvendor model, respectively. Syntetos et al. [27] not only demonstrate the superiority of deviating from the memoryless demand model, but also take the degree of intermittence into consideration and explore its connection with forecast accuracy through a numerical investigation. In sum, the above literature under a compound Erlang process all assume that the lead time is constant.

In spite of the many contributions to the material inventory management field, there are some shortcomings that need to be addressed. Firstly, the majority of research focuses on the service-oriented rather than cost-oriented inventory systems. For instance, Matheus and Gelder [15] demonstrate a simpler method to calculate the optimal reorder point subject to a compound Poisson demand due to the assumption of a target service level. Fung et al. [28] build a model of period-review with an order-up-to-level (*T, S*) policy for single-item inventory systems under compound Poisson demands and service-level constraints. Zhao [29] evaluates an Assemble-to-Order (ATO) system under compound Poisson demand with a batch ordering policy by means of service level, including the delivery lead-times and the order-based fill rates. Topan and Bayindir [30] explore the multi-item two-echelon material inventory system with compound Poisson demand and constraints on the aggregate mean response time, which is an indicator of a target service level. In addition, the proposed algorithm is not able to be implemented since it is complicated and does not fit bulk operation.

In response to these shortcomings, this paper is based on the electric power material demand data of Shanghai Electric Power Company and develops a method to determine the optimal order quantity with the policy, subject to the compound Erlang demand. Empirical analysis is conducted to verify the superiority of the proposed theoretical model in cost reductions, as compared to the Newsvendor model. This paper also proposes an approximation and a heuristic algorithm to overcome the inconvenience and low efficiency of existing algorithms in the real world.

#### 3. The Optimal Order Quantity Model with Compound Erlang Demand and (T,Q) Policy

##### 3.1. Problem Description and Model Assumptions

###### 3.1.1. Problem Description

This paper explores the inventory control of electric power materials with intermittent demand, which is a rather challenging task with tremendous cost implications for companies holding these inventories. The relevant cost mainly consists of holding cost incurred for each unit kept in stock and backordering cost arising from excess demand. Intermittent demand is modelled as a more realistic compound Erlang-k distribution which has an IFR. We consider a cost-oriented single echelon single-item inventory system where unfilled demand is backordered and lead time is deterministic. The stock level is controlled according to a periodic review policy, which means that an order with fixed-amount quantity is triggered every inspection interval* T*. The stock level is recorded as holding cost if exceeds the total demand sizes within the lead time and as backordering cost if not.

###### 3.1.2. Model Assumptions

This section shows relevant notations and assumptions of the inventory control model proposed in this paper.

We introduce the following notations: : lead time (constant) : order quantity : demand size (random variable) : probability distribution function of demand sizes : mean of nonzero demand sizes : mean demand arrival rate : the number of Erlang stages : inspection interval : inventory holding cost per SKU (Stock Keeping Unit) per unit of time : inventory backordering cost per SKU per unit of time : the complex number

For the proposed model, the following assumptions are made.

(1) A single echelon single-item inventory system, which means that only a particular item in a particular echelon of the supply chain is considered in the inventory system.

(2) A periodic review policy, where periodic inspections are performed every cycle time of T and a fixed-amount order quantity is subsequently generated. This policy means that organizations do nothing to control inventory within a cycle.

(3) The compound Erlang demand process, where demand arrivals are governed by Erlang distribution and demand sizes are governed by an arbitrary nonnegative probability distribution.

(3) Constant lead time.

(4) The total inventory cost consisting of inventory holding cost and backordering cost. Inventory holding cost is incurred for each unit kept in stock while backordering cost is due to the excess demand.

##### 3.2. Model Analysis

The model proposed in this paper presents demand as a compound Erlang process where demand arrivals are governed by an Erlang-k distribution and demand sizes are governed by an arbitrary nonnegative probability distribution. According to the above assumptions, the total expected inventory cost consists of the holding cost and backordering cost . Assuming that the probability of no-demand arrivals during the lead time is , the expected cost equals . When there are demand arrivals with a total size of during the lead time, the inventory holding cost constitutes the total cost when , while the backordering cost constitutes the total cost when . Thus, the expected total cost is given as follows:

Note that the probability distribution of demand size is the m-fold iterated convolution of . Besides, the probability of demand arrivals within the lead time in formula (1) can be derived based on the queuing and renewal theories (Saidane et al. [26]):

The objective of this paper is to derive the optimal order quantity , minimizing the expected total cost. Since is a convex function, it can be obtained by setting the derivative of the expected total cost with respect to equal to zero:

Proposition 1. *The optimal order quantity is the solution of *

*Proof. *The optimal order quantity is the solution of the derivative of the expected total cost with respect to which equals zero:which is equivalent toThusConsidering that , the optimal order quantity can be derived fromSince formula (8) is the infinite sum, this paper uses an upper bound and a lower bound to approximate the optimal order quantity . In order to derive and , we denote that

*Property 2. *, for all .

*Proof. *Since for any ,Thus :Since , .

Thus .

To sum up, .

*Property 3. *, , and are strictly increasing functions of .

*Proof. *Since , , and are sums of varied cumulative probability distribution functions which are strictly increasing, these three functions are strictly increasing.

*Property 4. *

*Proof. *When n tends to infinity, and tend to zero.

Assume that , , and are the solutions of , , and , respectively. Based on the above three properties, the following Proposition 5 is correct when n tends to infinity.

Proposition 5. *can be approximated by an upper bound and a lower bound based on , where is the solution of **and is the solution of *

##### 3.3. Basic Algorithm

Based the above analysis, the algorithm to compute the optimal order quantity is comprised the following steps.

*Step 0*. For a given arbitrarily small value , initialize

*Step 1*. Compute .

*Step 2* (i)If , go to Step 3. (ii)Otherwise do and go to Step 1.

*Step 3*. Utilize the derived n to solve (15) so as to obtain .

*Step 4*. Stop.

In order to help understand the basic algorithm, all above procedures are illustrated in the flow chart, i.e., Figure 1.