Mathematical Problems in Engineering

Volume 2016 (2016), Article ID 9328371, 9 pages

http://dx.doi.org/10.1155/2016/9328371

## Strategic Inventory Positioning in BOM with Multiple Parents Using ASR Lead Time

Department of Industrial Engineering, Ajou University, San 5, Woncheon-dong, Yeongtong-gu, Suwon 443-749, Republic of Korea

Received 30 October 2015; Revised 4 January 2016; Accepted 7 February 2016

Academic Editor: Jei-Zheng Wu

Copyright © 2016 Jingjing Jiang and Suk-Chul Rim. 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

In order to meet the lead time that the customers require, work-in-process inventory (WIPI) is necessary at almost every station in most make-to-order manufacturing. Depending on the station network configuration and lead time at each station, some of the WIPI do not contribute to reducing the manufacturing lead time of the final product at all. Therefore, it is important to identify the optimal set of stations to hold WIPI such that the total inventory holding cost is minimized, while the required due date for the final product is met. The authors have presented a model to determine the optimal position and quantity of WIPI for a given simple bill of material (S-BOM), in which any part in the BOM has only one immediate parent node. In this paper, we extend the previous study to the general BOM (G-BOM) in which parts in the BOM can have more than one immediate parent and present a new solution procedure using genetic algorithm.

#### 1. Introduction

Since 1964, extensive investigations have been conducted on the materials requirement planning (MRP), which ensures materials to be available to meet manufacturing and market demands. Utilizing master production schedule (MPS) and bill of material (BOM) as input data, MRP determines when and how many items to produce or purchase to meet the MPS. MRP has been widely used in most manufacturing industry for generating a detailed production schedule and purchase order for two decades since introduced.

However, due to the rapid development of technologies, wider choice of customers, and lower forecast accuracy, the performance of MRP became unsatisfactory. In 2011, Ptak and Smith [1] developed a new type of MRP, named as demand-driven MRP (DDMRP), which replaces the previous convention of safety stock with strategically replenished positions. As an innovative multiechelon pull methodology, DDMRP can plan inventory of materials, enable a company to build plans more closely to actual market demands, and promote better and quicker decisions and actions in the planning and execution. Traditional MRP focuses on answering for how much inventory to hold, and when to release order, while DDMRP focuses on answering for where to position the work-in-process inventory (WIPI).

The make-to-order (MTO) strategy makes the end product only when the customer places the order. The lead time for MTO is relatively longer compared to the make-to-stock (MTS) for the customer to get the product. In order to shorten the overall response time to the customer in MTO manufacturing, WIPI needs to be held in the station network. However, holding WIPI in some stations does not contribute to shortening the manufacturing lead time of the final product at all, only to increase the total inventory holding cost. So it is important to identify the optimal set of stations to place WIPI in the station network. We named this problem as the strategic inventory positioning (SIP) problem in our previous work (Rim et al. [2]). In the previous work, we address the SIP problem for the simple BOM (S-BOM) in which any part in the BOM has only one immediate parent part. The current study, however, is an extension or generalization of the previous work in that we address the SIP problem for the general BOM (G-BOM) in which some parts in the BOM have more than one immediate parent.

Only the SIP problem with BOM having a small number of parts can obtain the optimal solution by enumeration within a reasonable computation time. For S-BOM with items, SIP problem is to choose a set of zero-one decisions out of alternatives, of which one must be optimal. However, for G-BOM, if one part has parent parts, the number of alternatives needs to be multiplied by !. As more items have multiple parents in the BOM, it is impractical to enumerate all solutions due to the excessively long computation time.

Therefore the objective of this paper is to develop a new mathematical model and a new genetic algorithm to solve the SIP problem for G-BOM, in which some parts in the BOM have more than one immediate parent. Our goal is to find a solution method that can balance the solution quality with the computational feasibility. We will present genetic operators that will limit the search to a specific set of feasible solutions instead of exploring all possible solutions.

This paper is organized as follows: a review of the related literature is presented in Section 2. ASRLT in S-BOM and G-BOM is introduced in Section 3. Section 4 is dedicated to the problem description and model formulation of the new SIP problem in DDMRP. The process of building the genetic algorithm to solve the problem and the computational results using the proposed GA are given in Section 5. Conclusion and future works are given in Section 6.

#### 2. Literature Review

The inventory positioning problem has been studied by many researchers for past several years. In some previous studies, authors solved the problem of determining the optimal inventory position to increase the service fill rate and minimize the total inventory holding cost against a facility or full supply chains. In order to arrive the best service level, Whybark and Yang [3] proposed a controlled simulation experiment to decide where to place inventory. However, we determine the optimal inventory position depending on the replenishment model presented in DDMRP. Our paper has same targets including increasing service fill rate and minimizing total inventory holding cost with those in the previous literatures, but the method of calculating the lead time of the end product that the customer requires and the total inventory holding cost for all stations is different with those in previous research.

Many researchers apply Simpson’s model to solve the inventory positioning problems. Simpson Jr. [4] used the “*all or nothing*” policy to decide whether to place the inventory or not in some points in serial line system. He indicated that the service time is independent variable, and the total inventory holding cost is the objective function. The safety stock for time period is considered as the average inventory when he calculated the total inventory cost. He defined the safety stock as multiplying standard deviation of the demand for time period by the safety factor.

Graves and Willems [5–7] extended Simpson Jr. [4] model to assembly, distribution, and spanning tree network structures. They employed the periodic review base-stock-system model assuming no capacity constraints. In their paper, they considered the service time as a decision variable and minimizing the total inventory holding cost as the objective. They calculated the total inventory holding cost just as what Simpson Jr. [4] did but proposed DP algorithm.

Lesnaia [8] applied the Graves and Willems [6] model to the supply chain in a manufacturing firm. The difference is that Lesnaia considered the service time as a stochastic model instead of a deterministic model assumed by Graves and Willems. Lesnaia presented a general-network algorithm to manage the safety stock placement problem and determine the optimal service time for the path.

Magnanti et al. [9] solved the inventory positioning problem at production/assembly stages of components in an acyclic supply chain network structure which is not a spanning tree network structure. The service delivery time and inbound replenishment lead time are considered as decision variables and minimizing the total inventory cost is considered as the objective in all stages of the network. They built an efficient MIP formulation and employed a successive piecewise linear approximation approach.

Kaminsky and Kaya [10] developed an effective heuristic algorithm using linear programming modes to handle problems in supply chain including inventory positioning, short and reliable due-date quotation, and order sequencing. They defined two expected times as decision variables. One is that the safety stock of components stored at facility that are received from upstream facility for the production of type will last; another one is that the inventory of finished goods at facility required for the production of product type will last. The objective function consists of inventory costs, average lead time costs, and tardiness cost in a facility.

Inderfurth [11] and Inderfurth and Minner [12] applied the Simpson model to divergent and convergent systems in the network structure. The paper used a periodic review base-stock control policy and considered minimizing the inventory cost as the objective function and the service time as a decision variable.

Some researchers adopted the postponement concept in determining the inventory position in the supply chain such as Ioannou et al. [13]. This idea was initially introduced by Alderson [14] and then has been studied by Bucklin [15], Lee et al. [16], Davis [17], Feitzinger and Lee [18], and Ernst and Kamrad [19].

Rim et al. [2] stated the total inventory cost as the objective function and the strategic inventory position as decision variable that is different from the service time presented in previous studies. The paper used the new and important concept of ASRLT introduced by Ptak and Smith [1]. A mathematical model is built and GA method is proposed to determine the inventory position in BOM to minimize the total inventory cost while meeting the lead time required by customers. In this paper, we continue using the basic theory bought by our last paper (Rim et al. [2]). However, the BOM structure considered in last paper is simplified, so we generalize the BOM structure in this paper so that the calculation of ASRLT becomes more complicated. The new mathematical model and GA method are proposed in this paper so that the SIP problem in DDMRP is studied completely in aspect of BOM structure.

#### 3. ASRLT in S-BOM and G-BOM

In this chapter, we introduce Actively Synchronized Replenishment (ASR) lead time. ASR lead time is defined as the longest unprotected or unbuffered sequence in the BOM for a particular parent (Ptak and Smith [1]). Figure 1 from Rim et al. [2] shows that the manufacturing lead time of part 101, the end product of the BOM, is 2 days, given that three components of level 2 are all available. The longest path to calculate the cumulative lead time is 26 days from 101 through 201, 301,402, to 501P.